A conjectural formula for the minimum weight of Schubert codes was conjectured by Ghorpade in 2000. This was established by Xiang in 2008. In 2018, Ghorpade and Singh provided a new proof of this conjecture. Moreover, they also conjectured that the
minimum weight codewords of the Schubert codes \(C_{\alpha}(\ell, m)\) is given by the so-called Schubert decomposable codewords. We prove the validity of the conjecture for all Schubert varieties \(\Omega_{{\alpha}}(\ell, V_m)\) for all but finitely many values of \(q\).
Let \({\mathbb{F}}_q\) be a finite field with \(q\) elements, where \(q\) is a prime power. For positive integers \(n\)
and \(k\) with \(k \le n\), an \([n, k]_q\) linear code (or, simply an \([n, k]\)-code) \(C\) is a \(k\)-dimensional subspace of the vector space \({\mathbb{F}}_q^n\). The parameters \(n\) and \(k\) are called the length and the dimension of the code \(C\), respectively. Elements of a code are called codewords. The vector space, with usual coordinate system, can be endowed with a
metric, known as the Hamming metric, denoted by \(d_H\), where for any two elements \({\boldsymbol{a}} = (a_1, \dots, a_n) \in {\mathbb{F}}_q^n\) and \({\boldsymbol{b}} = (b_1, \dots, b_n) \in {\mathbb{F}}_q^n\), their Hamming distance is given by \[d_H ({\boldsymbol{a}}, {\boldsymbol{b}}) = |\{i : a_i \neq b_i\}|.\] Furthermore, the
Hamming distance naturally gives rise to the notion of Hamming weight of a codeword. Given an element \({\boldsymbol{c}} = (c_1, \dots, c_n) \in {\mathbb{F}}_q^n\), we define its Hamming weight, denoted by
\[w_H ({\boldsymbol{c}}) = |\{ i : c_i \neq 0\}|.\] We define the minimum distance of a code \(C\), denoted by \(d (C)\), as \[d(C) = \min \{d({\boldsymbol{x}}, {\boldsymbol{y}}) : {\boldsymbol{x}}, {\boldsymbol{y}} \in C, {\boldsymbol{x}} \neq {\boldsymbol{y}} \}.\] Due to the linearity of the code \(C\), it follows
that the minimum distance of a code \(C\) is equal to the minimum weight of \(C\), where the latter is defined as \[\min \{w_H ({\boldsymbol{c}}) :
{\boldsymbol{c}} \in C, {\boldsymbol{c}} \neq 0\}.\] In order to produce good codes, one often takes recourse to algebraic varieties over finite fields and codes that occur from them. One way to construct codes from algebraic varieties is to use the
notion of a projective system. We omit a detailed discussion of projective systems and codes here, but refer the reader to [1].
The codes from Schubert varieties, also known as Schubert codes, were first introduced in [2] by Ghorpade and Lachaud. Let \({\alpha}=
({\alpha}_1, \dots, {\alpha}_{\ell}) \in I(\ell, m)\), as defined in 2 , \(V_m\) a vector space of dimension \(m\) over \({\mathbb{F}}_q\) with an ordered basis \(\mathcal{B}\) and \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) be the Schubert subvariety of the Grassmannian \(G(\ell, V_m)\) embedded in a nondegenerate manner in the projective space \({\mathbb{P}}^{k_{\alpha}- 1}\) as described in Subsection 2.2. It turns out the length
\(n_{\alpha}\), dimension \(k_{\alpha}\), and the minimum distance/weight \(d_{\alpha}\) of the code \(C_{{\alpha}} (\ell, V_m,
\mathcal{B})\) arising from the Schubert subvariety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) are given by the following formulas:
\(n_{\alpha}= |C_{{\alpha}} (\ell, V_m, \mathcal{B})|\). We refer to [3] for combinatorial formulas describing \(n_{\alpha}\). We also discuss this matter at length in Subsection 3.1 later.
\(d_{\alpha}= n_{\alpha}- \max \{|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| : \Pi_{\alpha}\;\text{is a hyperplane of {\mathbb{P}}^{k_{\alpha}- 1}} \} = q^{\delta ({\alpha})}\), where \(\delta ({\alpha}) = ({\alpha}_1 + \dots + {\alpha}_\ell) - \frac{\ell (\ell + 1)}{2}\). This formula was first conjectured in [2], and
subsequently proved in [4] and [5] in the case \(\ell
= 2\), and in full generality in [6]. An independent proof for this formula was also given in [7].
We note that when \({\alpha}= (m - \ell + 1, \dots, m)\), the Schubert subvariety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) equals the Grassmannian \(G(\ell,
V_m)\). In this particular case, all the above formulas are known: \[n_{\alpha}= {m \brack \ell}_q, \;k_{\alpha}= {m \choose \ell} \;\text{and} \;d_{\alpha}= q^{\ell (m - \ell)}.\] While the first two formulas are
classical, the minimum distances of this particular Schubert code, namely the Grassmann code, were determined by Nogin [8]. Moreover, Nogin also provided a complete
classification of the minimum-weight codewords of Grassmann codes. Even though, in general, the minimum weight of the Schubert codes is known, a classification of all the minimum weight codewords is yet to be done. In [7], the authors conjectured a classification of all the minimum weight codewords of Schubert codes (cf. Conjecture 1).
Their conjecture states that the minimum-weight codewords of Schubert codes are given by Schubert decomposable hyperplanes (cf. Definition 1). Ghorpade and Singh proved the conjecture
in the affirmative in the case when \(\ell = 2\). In this paper, we show that the conjecture is true for all positive integers \(\ell\), when \(q\) is
sufficiently large. In achieving this, we also give an alternative proof of the formula for \(d_{\alpha}\) as mentioned above.
This paper is organized as follows: In Section 2, we gather all the preliminary definitions and results that are needed to prove our main result. This includes a detailed discussion on Grassmann varieties and their Schubert subvarieties. In Section 3,
we provide several combinatorial decompositions of Schubert subvarieties and discussions on the enumerations on the points of Schubert subvarieties over finite fields. Finally, in Section 4, we prove our main result which is given in Theorem 15.
In this section, we recall some of the properties of Grassmannians and their Schubert subvarieties. Our main goal is to record the well-known properties of these objects that will be used later in this paper. None of the results in this section are new.
They can be found in [9]–[12], and in modern texts such as [12]. Moreover, we borrow some definitions and results in coding theory from several sources [2], [4], [6]–[8], [13], [14]. We also refer to [15] and the references therein for a more detailed understanding of the material covered in this section.
2.1 Grassmannian, Plücker projective space, and Plücker embedding↩︎
Fix integers \(\ell, m\) satisfying \(1 \le \ell \le m\) and an \(m\)-dimensional vector space \(V_m\) over a field \(\mathbb{F}\) with an ordered basis \(\mathcal{B} = \{v_1, \dots, v_m\}\) of \(V_m\). The Grassmannian of all \(\ell\)-dimensional subspaces of \(V_m\), denoted by \(G (\ell, V_m)\) is defined as \[G (\ell, V_m) :=\{ L \subset V_m : L
\;\text{is a subspace of} \;V_m, \;\dim L = \ell\}.\] Let us now impose a geometric structure on \(G(\ell, V_m)\) via the so-called Plücker embedding that is given by: \[\label{pl1} \pi_{\ell, m} : G(\ell, V_m) \to {\mathbb{P}}\left(\bigwedge^\ell V_m\right) \;\;\text{given by} \;\;L \mapsto [\omega_1 \wedge \cdots \wedge \omega_\ell],\tag{1}\] where \(\{\omega_1, \dots, \omega_\ell\}\) is a basis of \(L\). It is well-known that the map is well-defined, that is, it is independent of the choice of basis for the elements of \(G(\ell, V_m)\). The projective space \({\mathbb{P}}\left(\bigwedge^\ell V_m\right)\) is known as the Plücker projective space. Let \[\label{ilm}I(\ell, m) := \{{\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell) \in \mathbb{Z}^\ell : 1 \le {\alpha}_1 < \cdots < {\alpha}_\ell \le m\}.\tag{2}\] Note that \(\bigwedge^{\ell}
V_m\) is a \(k = {m \choose \ell}\)-dimensional vector space over \(\mathbb{F}\). Moreover, a fixed ordering of elements of \(I(\ell,m)\) gives an
ordered basis \(\{v_{\alpha}= v_{{\alpha}_1} \wedge \cdots \wedge v_{{\alpha}_\ell} : {\alpha}\in I(\ell, m)\}\) of \(\bigwedge^{\ell} V_m\). This ordered basis, in turn, helps us in
introducing the homogeneous coordinates of the Plücker projective space. In particular, if \(L \in G(\ell, V_m)\) and \(A_L = (a_{ij})\) is an \(\ell \times
m\) matrix with entries in \({\mathbb{F}}\) such that the row-space of \(A_L\) is \(L\), then \[\label{homcoor} \pi_{\ell, m} (L) = [p_{\alpha}(A_L)]_{{\alpha}\in I(\ell, m)}\tag{3}\] where \(p_{{\alpha}} (A_L)\) is the \({\alpha}\)-th minor of
\(A_L\). As mentioned above, it is known (cf. [12]) that \(\pi_{\ell, m} (L)\) is independent of the
choice of \(A_L\) and consequently the map \(\pi_{\ell, m}\) is well-defined. Moreover, the map \(\pi_{\ell, m}\) is injective [12]. We remark that the rows of \(A_L\) above are written with respect to the fixed basis \(\mathcal{B}= \{v_1, \dots,
v_m\}\) of \(V_m\). However, to our advantage, it turns out that changing the basis of \(V_m\), does not affect the situation drastically. More precisely, a change of basis of \(V_m\) induces a collineation of \({\mathbb{P}}(\bigwedge^\ell V_m)\) taking \(G(\ell, V_m)\) to itself. When \({\mathbb{F}}\) is
an algebraically closed field, it can be shown that \(G(\ell, V_m)\) is given by the solutions to a system of some homogeneous quadratic equations in \(X_{\alpha}\)-s, known as the Plücker
relations. We refer to [12] for a complete proof of this fact. Consequently, the subset \(G(\ell, V_m)\) can be regarded as a
projective algebraic variety when working over an algebraically closed field.
We keep working with the fixed ordered basis \(\mathcal{B}= \{v_1, \dots, v_m\}\) of \(V_m\), as mentioned above. For every \(L \in G(\ell, V_m)\), We may
choose a basis of \(L\) in a way so that \(A_L\) is in right-row-reduced-echelon form, which is uniquely determined by \(L\). That is, \(L\) can be uniquely represented by an \(\ell \times m\) matrix \(M_{\mathcal{B}}(L)\) satisfying
the rows of \(M_{\mathcal{B}}(L)\) are elements of \(L\) written with respect to the basis \(\mathcal{B}\) of \(V_m\).
the row-space of \(M_{\mathcal{B}}(L)\) is \(L\),
for each \(i = 1, \dots, \ell\) the last non-zero entry of the \(i\)-th row, called the pivot of \(i\)-th row, is equal to \(1\),
the last non-zero entry of \((i+1)\)-st row appears to the right of the last nonzero entry of the \(i\)-th row,
all the entries above and below of a pivot are \(0\).
In particular, we have a natural one-to-one correspondence, \[\label{em} G(\ell, V_m) \longleftrightarrow \mathcal{M}(\ell, m) \;\;\text{given by} \;\;L \longleftrightarrow
M_{\mathcal{B}}(L),\tag{4}\] where \(\mathcal{M}(\ell, m)\) is the set of all \(\ell \times m\) matrices in right-row-reduced echelon form with entries in \({\mathbb{F}}\). Moreover, with respect to a fixed basis \(\mathcal{B}\) of \(V_m\), the map \(\pi_{\ell, m}\) in 1 can be rewritten as \[\label{pl2} \pi_{\ell, m}: G (\ell, V_m) \to {\mathbb{P}}\left(\bigwedge^\ell V_m\right) \;\;\text{as} \;\;\pi_{\ell, m} (L) =
[p_{\alpha}(M_{\mathcal{B}}(L))]_{{\alpha}\in I(\ell, m)}.\tag{5}\] This description will be used later in this article.
The Grassmannian \(G(\ell, V_m)\) contains a special class of subvarieties, known as the Schubert subvarieties, which are the objects of central interest in this article. For \({\alpha}\in I(\ell, m)\), the \({\alpha}\)-th Schubert cell, with respect to the basis \(\mathcal{B}=\{v_1, \dots, v_m\}\), denoted by \(C_{{\alpha}}(\ell, V_m, \mathcal{B})\), is defined as, \[C_{{\alpha}}(\ell, V_m, \mathcal{B}) := \{L \in G(\ell, V_m) : \;\text{the pivots of} \;M_{\mathcal{B}}(L) \;\text{are on the columns}
\;{\alpha}_1, \dots, {\alpha}_\ell \}.\] Note that \(C_{\alpha}(\ell, V_m, \mathcal{B})\) is identified with an affine space of dimension \(\delta ({\alpha})\) over \({\mathbb{F}}\), where \[\delta ({\alpha}) = ({\alpha}_1 + \dots + {\alpha}_\ell) - \frac{\ell (\ell+1)}{2}.\] We define the partial order, also known as the Bruhat order, on the elements of
\(I(\ell, m)\) as follows: For \({\alpha}= ({\alpha}_1, \dots, {\alpha}_{\ell}), \beta = (\beta_1, \dots, \beta_{\ell}) \in I(\ell, m)\), we say that \[{\alpha}\le
\beta \iff {\alpha}_i \le \beta_i \;\;\;\text{for all} \;i \in \{1, \dots, \ell\}.\] For \({\alpha}\in I(\ell, m)\), define \[\Delta ({\alpha}) = \{\beta \in I(\ell, m) : \beta \not\le
{\alpha}\}, \;\nabla ({\alpha}) = I(\ell, m) \setminus \Delta ({\alpha}), \;\text{and} \;k_{\alpha}= |\nabla({\alpha})|.\] The \({\alpha}\)-th Schubert subvariety in\(G(\ell,
V_m)\) with respect to the basis \(\mathcal{B}\), denoted by \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\), is defined as \[\label{Sch}
\Omega_{\alpha}(\ell, V_m, \mathcal{B}):= \bigcup_{\beta \in \nabla({\alpha})} C_{\beta} (\ell, V_m, \mathcal{B}).\tag{6}\] Indeed, the union mentioned above is clearly a disjoint union of Schubert cells. Evidently, if \({\alpha}= (m - \ell + 1, \dots, m)\), then \(\Omega_{\alpha}(\ell, V_m, \mathcal{B}) = G(\ell, V_m)\). For \(i= 1, \dots, m\), let \(V_i\) be the subspace of \(V_m\) spanned by \(\{v_1, \dots, v_i\}\). We see that for every \({\alpha}= ({\alpha}_1, \dots,
{\alpha}_\ell) \in I(\ell, m)\), there is a partial flag of subspaces of \(V_m\) given by \[0 \subset V_{{\alpha}_1} \subset \cdots \subset V_{{\alpha}_\ell} \subset V_m \;\;\;\text{with}
\;\;\;\dim V_{{\alpha}_i} = {\alpha}_i \;\;\text{for all} \;i =1, \dots, \ell.\] Moreover, it follows rather trivially that \[L \in C_{{\alpha}} (\ell, V_m, \mathcal{B}) \iff \dim L \cap V_{{\alpha}_j} = j \;\; \text{for
every} \;j = 1, \dots, \ell,\] while \[L \in \Omega_{{\alpha}} (\ell, V_m, \mathcal{B}) \iff \dim L \cap V_{{\alpha}_j} \ge j \;\; \text{for every} \;j = 1, \dots, \ell.\] In literature, several authors define a
Schubert subvariety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) in terms of the above equivalence condition. The Plücker embedding \(\pi_{\ell, m}\) restricted to \(\Omega_{\alpha}(\ell, V_m, \mathcal{B})\) allows us to view the Schubert variety as a subset of the projective space \({\mathbb{P}}\left(\bigwedge^{\ell} V_m\right)\). As mentioned above, with
respect to the basis \(\mathcal{B}\) and a fixed ordering of the elements of \(I(\ell, m)\), the projective space \({\mathbb{P}}^{k-1} =
{\mathbb{P}}\left(\bigwedge^{\ell} V_m\right)\) is endowed with the homogeneous coordinates \((X_{{\alpha}})_{{\alpha}\in I(\ell, m)}\). We define the projective linear subspace \({\mathbb{P}}^{k_{\alpha}- 1}\) of \({\mathbb{P}}\left(\bigwedge^{\ell} V_m\right)\) by \[\label{pkalpha}
{\mathbb{P}}^{k_{\alpha}- 1} := \left\{ P \in {\mathbb{P}}\left(\bigwedge^{\ell} V_m\right) : X_\beta (P) = 0 \;\text{for all} \;\beta \in \Delta ({\alpha})\right\}.\tag{7}\] It is well-known that \[\label{commd1} \pi_{\ell, m} (\Omega_{\alpha}(\ell, V_m, \mathcal{B})) = G(\ell, V_m) \cap {\mathbb{P}}^{k_{\alpha}- 1},\tag{8}\] where \({\mathbb{P}}^{k_{\alpha}- 1}\) is as defined
above. As with the Grassmann varieties, we will denote by \(\Omega_{\alpha}(\ell, V_m, \mathcal{B})\), the Plücker embedding of \(\Omega_{\alpha}(\ell, V_m, \mathcal{B})\) in \({\mathbb{P}}^{k_{\alpha}- 1}\). We see that the \({\alpha}\)-th Schubert variety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) can be geometrically understood as
a linear section of the Grassmann variety \(G(\ell, V_m)\) given by some coordinate hyperplanes of the Plücker space.
2.3 Hyperplanes, their decomposability, and Schubert decomposability↩︎
One of our main goals in this article is to study the maximum number of points lying on a hyperplane section of a Schubert variety \(\Omega_{\alpha}(\ell, V_m, \mathcal{B})\) defined over a finite field. That
is, we want to determine the quantity \[\max \{|\Pi_{\alpha}\cap \Omega_{\alpha}(\ell, V_m, \mathcal{B})|: \Pi_{\alpha}\;\text{is a hyperplane of} \;{\mathbb{P}}^{k_{\alpha}- 1} \},\] when all the subsets mentioned above
are defined over a finite field \({\mathbb{F}}_q\). To this end, it is imperative to first understand the algebraic and geometric properties of hyperplanes in the Plücker space. We will briefly recall the definitions and
the properties of them in this subsection, but refer the reader to [15] and the references therein for more leisurely reading.
The hyperplanes of the Plücker space \({\mathbb{P}}(\bigwedge^\ell V_m)\) can be represented in two ways. On one hand, they are given by elements of the dual vector space \((\bigwedge^\ell
V_m)^*\) which is known to be isomorphic to \((\bigwedge^{m -\ell} V_m)\). Thus a hyperplane of \({\mathbb{P}}(\bigwedge^\ell V_m)\) can be defined by an element \(F\), given by \[F = \sum_{{\alpha}\in I(m - \ell, m)} c_{\alpha}v_{{\alpha}_1} \wedge \cdots\wedge v_{{\alpha}_{m - \ell}} \in \bigwedge^{m-\ell} V_m.\] As usual, a point \(L \in G(\ell, V_m)\) lies on the hyperplane given by \(F\) if and only if \(F(L) =0\). Here \[F(L) = \sum_{{\alpha}\in I(m - \ell, m)}
c_{\alpha}v_{{\alpha}_1} \wedge \cdots\wedge v_{{\alpha}_{m - \ell}} \wedge (w_1 \wedge \dots \wedge w_\ell),\] where \(\{w_1, \dots, w_\ell\}\) is an ordered basis of \(L\). On the
other hand, we may also describe a hyperplane of \({\mathbb{P}}(\bigwedge^\ell V_m)\) geometrically in terms of the homogeneous coordinates \(X_{\alpha}\) as mentioned earlier. Here, we may
represent a hyperplane as the vanishing set of a homogeneous linear polynomial \[\sum_{{\alpha}\in I(\ell, m)} c_{\alpha}X_{\alpha}.\] It is understood that, for \({\alpha}= ({\alpha}_1, \dots,
{\alpha}_\ell)\) the coordinate hyperplane given by \(X_{\alpha}\) corresponds to the hyperplane given by the element \(v_{{\alpha}^{\mathsf{C}}} = v_{{\alpha}'_1} \wedge \cdots \wedge
v_{{\alpha}'_{m-\ell}} \in \bigwedge^{m-\ell} V_m\), where \(\{{\alpha}'_1, \dots, {\alpha}'_{m-\ell}\} \cup \{{\alpha}_1, \dots, {\alpha}_\ell\} = \{1, \dots, m\}\). This correspondence naturally extends
to the following identification \[\label{hypid} \sum_{{\alpha}\in I (\ell, m)} c_{\alpha}X_{{\alpha}} \longleftrightarrow \sum_{{\alpha}\in I(\ell, m)}
c_{\alpha}v_{{\alpha}^{\mathsf{C}}},\tag{9}\] where for each \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell) \in I(\ell, m)\), the element \({\alpha}^{\mathsf{C}} = ({\alpha}'_1,
\dots, {\alpha}'_{m-\ell}) \in I(m - \ell, m)\) satisfying \(\{{\alpha}'_1, \dots, {\alpha}'_{m-\ell}\} \cup \{{\alpha}_1, \dots, {\alpha}_\ell\} = \{1, \dots, m\}\). The non-degeneracy of the
Grassmannian \(G(\ell, V_m)\) in \({\mathbb{P}}(\bigwedge^\ell V_m)\) implies that if \(\sum_{{\alpha}\in I(\ell, m)} c_{\alpha}X_{\alpha}(L) =0\) for all
\(L \in G(\ell, V_m)\), then \(c_{\alpha}= 0\) for all \({\alpha}\in I(\ell, m)\).
Let us now consider the \({\alpha}\)-th Schubert subvariety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). We recall that \(\Omega_{\alpha}(\ell, V_m,
\mathcal{B}) = G(\ell, V_m) \cap {\mathbb{P}}^{k_{\alpha}- 1}\), where \({\mathbb{P}}^{k_{{\alpha}} - 1}\) is the linear subspace of \({\mathbb{P}}^{k-1}\) as given in 7 . Take a hyperplane \(\Pi\) of \({\mathbb{P}}(\bigwedge^{\ell} V_m)\) given by \(F (X_{\alpha}: {\alpha}\in I(\ell, m)) = \sum_{\beta \in
I(\ell, m)} c_\beta X_\beta\). We may write \[F = F_{\Delta ({\alpha})} + F_{\nabla ({\alpha})}, \;\;\text{where} \;\;F_{\Delta ({\alpha})} = \sum_{\beta \in \Delta ({\alpha})} c_\beta X_\beta \;\;\text{and} \;\;F_{\nabla
({\alpha})} = \sum_{\beta \in \nabla ({\alpha})} c_\beta X_\beta.\] We see that a hyperplane \(\Pi\) restricts to a hyperplane \(\Pi_{\alpha}\) in \({\mathbb{P}}^{k_{\alpha}- 1}\) if \(F_{\nabla ({\alpha})} \neq 0\). Indeed the restricted hyperplane \(\Pi_{\alpha}\) is defined as the set of zeroes of \(F_{\nabla ({\alpha})}\) in \({\mathbb{P}}^{k_{\alpha}- 1}\). Moreover, it is well-known (cf. [12]) that
the \({\alpha}\)-th Schubert variety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) of \(G(\ell, V_m)\) is also a non-degenerate subset of \({\mathbb{P}}^{k_{\alpha}- 1}\). Consequently, the hyperplane \(\Pi\) described above contains the Schubert variety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\)
if and only if \(F_{\nabla ({\alpha})} = 0\). While a hyperplane of the Plücker space restricts uniquely to a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\), a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) can be extended to a hyperplane of the Plücker space in several ways. That is, if \(\Pi_{\alpha}\) is a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) given by the vanishing set of \(F_{\alpha}= \sum_{\beta \in \nabla ({\alpha})} c_{\alpha}X_{\alpha}\), then one can construct a hyperplane \(\Pi\) of \({\mathbb{P}}(\bigwedge^\ell V_m)\) given by \(F = F_{\alpha}+ \sum_{\gamma \in \Delta ({\alpha})} c_\gamma X_\gamma\) such that \(\Pi \cap {\mathbb{P}}^{k_{\alpha}- 1} = \Pi_{\alpha}\). One readily checks that the choice of \(\Pi\) mentioned above is not unique. A few remarks are in order and we summarize them below.
Remark 1.
Our notation \(C_{\alpha}(\ell, V_m, \mathcal{B})\) to define the \({\alpha}\)-th Schubert variety, emphasizing its dependence on the basis \(\mathcal{B}\), while letting \(G(\ell, V_m)\) denote the Grassmannian independent of basis might lead to some confusion. As mentioned earlier, we see that when we change the basis of \(V_m\), it only induces a projective linear isomorphism of the Plücker space that fixes \(G(\ell, V_m)\) as a set. It is understood, of course, that the homogeneous coordinates of the Plücker
space get changed. However, for two bases \(\mathcal{B}\) and \(\mathcal{B}'\) of \(V_m\), the sets \(\Omega_{\alpha}(\ell,
V_m, \mathcal{B})\) and \(\Omega_{\alpha}(\ell, V_m, \mathcal{B}')\) can be different as subsets.
As before, let \(\mathcal{B}=\{v_1, \dots, v_m\}\) be a fixed basis of \(V_m\) and for each \(i=1, \dots, m\), let \(V_i\) denote the subspace spanned by \(\{v_1, \dots, v_i\}\). It is easily seen that \(\Omega_{\alpha}(\ell, V_m, \mathcal{B}) = \Omega_{\alpha}(\ell,
V_{{\alpha}_\ell}, \mathcal{B}_{{\alpha}_\ell})\) where \(\mathcal{B}_{{\alpha}_\ell} = \{v_1, \dots, v_{{\alpha}_\ell}\}\). In what follows, we shall use this identity. Indeed, this means we can view \(\Omega_{{\alpha}} (\ell, V_m, \mathcal{B})\) as a subset of \(G(\ell, V_{{\alpha}_\ell})\). From now on, when we mention the Schubert subvariety \(\Omega_{\alpha}(\ell,
V_m, \mathcal{B})\), we will always assume that \({\alpha}_\ell = m\) and that \(V_{{\alpha}_\ell} = V_m\).
A nonzero element \(z \in \bigwedge^{m- \ell} V_m\) is said to be decomposable if there exist \(w_1, \dots, w_{m-\ell} \in V_m\) such that \(z = w_1
\wedge \cdots \wedge w_{m-\ell}\). A hyperplane of \({\mathbb{P}}\left(\bigwedge^\ell V_m\right)\) is said to be decomposable if it is given by a decomposable element of \(\bigwedge^{m- \ell} V_m\). Thus a hyperplane \(\Pi\) of \({{\mathbb{P}}\left(\bigwedge^{\ell} V_m\right)}\) given by the equation \(\displaystyle{\sum_{{\alpha}\in I(\ell, m)}} c_{\alpha}X_{\alpha}\) is decomposable if \(\displaystyle{\sum_{{\alpha}\in I(\ell, m)} c_{\alpha}v_{{\alpha}^{\mathsf{C}}}}\) is a decomposable
element of \(\bigwedge^{m - \ell} V_m\).
Remark 2. For a nonzero element \(z \in \bigwedge^{m - \ell} V_m\), we define \({\mathcal{V}}(z) = \{x \in V_m : z \wedge x = 0\}\). We have
[11]\(z\) is decomposable if and only if \(\dim {\mathcal{V}}(z) = m -
\ell\).
We now recall a suitable generalization of the notion of decomposable hyperplanes in the setting of Schubert subvarieties as given in [7].
Definition 1 (Schubert Decomposability, [7]). Suppose \({\alpha}\in I(\ell, m)\) and \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) be the \({\alpha}\)-th Schubert subvariety of \(G(\ell, V_m)\) with respect to the fixed basis \(\{v_1, \dots, v_m\}\). We divide \({\alpha}\) into blocks of consecutive components* as follows: \[({\alpha}_1, \dots, {\alpha}_\ell) = ({\alpha}_1, \dots,
{\alpha}_{p_1}, {\alpha}_{p_1 + 1}, \dots, {\alpha}_{p_2}, \dots, {\alpha}_{p_u +1}, \dots, {\alpha}_\ell),\] where \(u, p_0, \dots, p_{u+1}\) are unique integers determined by \({\alpha}\) satisfying*
\({\alpha}_{p_i + 1} - {\alpha}_{p_i} \ge 2\) for \(1 \le i \le u\), and
\({\alpha}_{p_i - j} = {\alpha}_{p_i} - j\) for \(1 \le i \le u+1\) and \(1 \le j <p_i - p_{i-1}\).
This allows us to consider the refined partial flag* of subspaces associated with \({\alpha}\). That is, we now consider the partial flag: \[0 \subset V_{{\alpha}_{p_1}} \subset \cdots
\subset V_{{\alpha}_{p_u}} \subset V_m \;\;\;\text{where} \;\;V_{{\alpha}_i} = \text{span} \{v_1, \dots, v_{{\alpha}_i}\} \;\text{for} \;i=p_1, \dots, p_u.\]*
An element \(z \in \bigwedge^{m-\ell} V_m\) is said to be Schubert decomposable with respect to* \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) if \(z\) is decomposable and \(\dim ({\mathcal{V}}(z) \cap V_{{\alpha}_{p_i}}) = {\alpha}_{p_i} - p_i\) for all \(i = 1, \dots, u\). A hyperplane \(\Pi_{\alpha}\) of \(\mathbb{P}^{k_{{\alpha}}-1}\) given by \(F_{\alpha}\) is said to be a Schubert decomposable hyperplane if there exists a Schubert
decomposable element \(F \in \bigwedge^{m-\ell} V_m\) such that \(F_{\nabla ({\alpha})} = F_{\alpha}\) and the hyperplane \(\Pi\) defined by \(F\) in \({\mathbb{P}}^{k-1}\) restricts to \(\Pi_{\alpha}\) in \({\mathbb{P}}^{k_{\alpha}- 1}\).*
3 Schubert varieties over finite fields: a decomposition into
“smaller" Schubert varieties, and a combinatorial upper bound↩︎
From now onwards, we will denote by \({\mathbb{F}}_q\) a finite field with \(q\) elements where \(q\) is a fixed prime power. In this section, we will
focus our attention to the study of Grassmannians and their Schubert subvarieties over \({\mathbb{F}}_q\). One of our main goals in this paper is to answer the following question:
Question 3. Let \(\ell, m\) be positive integers satisfying \(1 \le \ell \le m\). Let \({\alpha}\in I(\ell, m)\), \(V_{m}\) be an \(m\)-dimensional vector space over \({\mathbb{F}}_q\) with a fixed ordered basis \(\mathcal{B}\), \(G(\ell, V_m)\) the Grassmannian of all \(\ell\) dimensional subspaces of \(V_m\) and \(\Omega_{\alpha}(\ell, V_m, \mathcal{B})\)
be the Schubert subvariety of \(G(\ell, V_m)\) with respect to \(\mathcal{B}\).
Determine \(e_{\alpha}(\ell, m) := \max \{ |\Pi \cap \Omega_{{\alpha}} (\ell, V_m, \mathcal{B})| : \Pi \;\text{is a hyperplane of} \;{\mathbb{P}}^{k_{\alpha}-1}\}\).
Classify all the hyperplanes \(\Pi\) of \({\mathbb{P}}^{k_{\alpha}- 1}\) such that \(|\Pi \cap \Omega_{{\alpha}} (\ell, V_m, \mathcal{B})| =
e_{\alpha}(\ell, m)\).
Indeed, the quantity \(e_{\alpha}(\ell, m)\) is independent of the choice of basis of \(V_m\), and consequently, there is no scope for ambiguity with a basis-independent notation being
used for the same. In the special case when \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})= G(\ell, V_m)\) the answer to Question 3 is known due to Nogin [8]. The answer to Question 3 (a) is known thanks to [6] and [7]. In fact, we have the following:
Theorem 4. [6], [7]We have \(e_{\alpha}(\ell, m) = |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\delta ({\alpha})}\).
As mentioned in the Introduction, the above theorem was proved in the special case when \(\ell =2\) in [4] and independently in [5]. We have the following conjecture [7] which we will refer to as the minimum
weight codewords conjecture or MWCC in this article.
Conjecture 1 (MWCC, [7]). If a hyperplane \(\Pi\) satisfies \(|\Pi \cap
\Omega_{{\alpha}} (\ell, V_m, \mathcal{B})| = e_{\alpha}(\ell, m)\), then \(\Pi\) is Schubert decomposable.
Remark 5. The conjecture is known to be true in the following special cases:
[7] If \(\Pi\) is decomposable, that is, \(\Pi\) extends to a decomposable hyperplane in
\({\mathbb{P}}(\bigwedge^\ell V_m)\);
[7] If \({\alpha}_1, \dots, {\alpha}_\ell\) are completely non-consecutive;
\({\alpha}_1, \dots, {\alpha}_\ell\) are consecutive: in fact, from the discussion in Remark 1, the corresponding Schubert variety is
a Grassmannian \(G(\ell, V_m)\);
It is also known [7] that if \(\Pi\) is Schubert decomposable, then \(|\Pi \cap \Omega_{{\alpha}} (\ell,
V_m, \mathcal{B})| = e_{\alpha}(\ell, m)\).
3.1 Enumeration of Schubert varieties over finite fields↩︎
The first question one would encounter is the following: How many points are there on a Schubert variety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) defined over a finite field \({\mathbb{F}}_q\)? Indeed, if \({\alpha}= (m - \ell+1, \dots, m)\), then \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})= G(\ell, V_m)\) and its cardinality is given
by the well-known Gaussian Binomial coefficient\({m \brack\ell}_q\) defined as follows: \[{m \brack \ell}_q := \frac{(q^m -1) \cdots (q^m - q^{\ell -1})}{(q^\ell -1) \cdots (q^\ell -
q^{\ell -1})}.\] The Gaussian binomials are well-studied in mathematics, especially in combinatorics. It is beyond the scope of the article to discuss several nice properties of them, but we refer the readers to [9] for a more leisurely reading on this topic. We now look at the general cases of Schubert varieties \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). For this,
let us look at the equation 6 . From this, we already know the following: \[\label{schen} |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| = \sum_{\beta \in \nabla ({\alpha})}
|C_\beta (\ell, V_m, \mathcal{B})| = \sum_{\beta \in \nabla({\alpha})} q^{\delta(\beta)} \;\;\text{where} \;\;\delta(\beta) = \sum_{i=1}^\ell \beta_i - \frac{\ell (\ell + 1)}{2}.\tag{10}\] The last equality above follows from 6 and the well-known fact that \(C_\beta (\ell, V_m, \mathcal{B})\) is an affine space of dimension \(\delta (\beta)\) over \({\mathbb{F}}_q\). This fact is easy to verify directly from looking at the matrices in right-row-reduced-echelon form that belong to \(C_\beta (\ell, V_m, \mathcal{B})\). It is difficult to
trace back in the literature to find the first proof of the same, but we refer to [3] and the references therein. To that end, Now we present a rather weak upper bound
for the number of points on \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). We begin with a Lemma.
Lemma 1. For \(\ell \ge 2\) and \({\alpha}\in I(\ell, m)\), we have \[\sum_{\beta \in \nabla ({\alpha})} q^{\beta_1 + \dots + \beta_\ell}
< \left(\frac{q}{q-1}\right)^\ell q^{{\alpha}_1 + \dots + {\alpha}_\ell}.\]
Proposition 6. We have \(|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| < \left(\frac{q}{q-1}\right)^\ell q^{\delta({\alpha})}\).
Proof. Follows directly from Lemma 1, 10 , and the definition \(\delta ({\alpha}) = {\alpha}_1 + \dots + {\alpha}_\ell
- \frac{\ell(\ell+1)}{2}\) for all \({\alpha}\in I(\ell, m)\). ◻
Remark 7.
In order to completely understand the number of points on \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\), one may write: \[\label{schen1}
|\Omega_{\alpha}(\ell, V_m, \mathcal{B})| = \sum_{i=0}^{\delta ({\alpha})} a_i q^i.\qquad{(1)}\] The integers \(a_i\) equals the number of Schubert cells of dimension \(i\) in
\(\Omega_{\alpha}(\ell, V_m, \mathcal{B})\). It is in general not difficult to determine these quantities. However, it is conceivable that determination of \(a_i\)-s will give rise to a
better bound for \(|\Omega_\alpha (\ell, V_{m}, \mathcal{B})|\).
A bound analogous to Proposition 6 for Gaussian binomials can be found in the literature. For example, it follows from [16] that \({m \brack \ell}_q < \frac{q^2}{q^2 - q - 1} q^{\ell (m-\ell)}\). Indeed, in the special case when the concerned Schubert subvariety is a Grassmannian, we see that the above
bound is stronger in general, while they coincide with our bound when \(q = 2\). We believe that the bound we presented above can be improved and leave this for future investigations.
3.2 A combinatorial bound on Schubert subvarieties↩︎
As noted above, it is not always easy to explicitly determine the coefficients \(a_i\) as presented in equation ?? . Here, our goal is to bypass the determination of these quantities while still obtaining some effective
combinatorial identities on the number of points on Schubert varieties, which are of independent interest and facilitate calculations. We stick to our basic definitions of \(\ell, m, {\alpha}\) and work with a fixed ordered
basis \(\mathcal{B}\) of \(V_m\). First, we need to introduce some more notations.
Definition 2. Suppose \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell) \in I(\ell, m)\) and \({\alpha}_1, \dots, {\alpha}_\ell\) are not consecutive integers. We
define:
\(k := \max \{j : a_{j+1} - a_{j} \ge 2\}\). Since \({\alpha}_1, \dots, {\alpha}_\ell\) are not consecutive integers, we have \(1 \le k \le \ell -
1\).
\({\alpha}' := ({\alpha}_1', \dots, {\alpha}_\ell')\) where \[{\alpha}_i' = \begin{cases} {\alpha}_i \;\;\;\;\;\;\;\;\;\text{if} \;\;\;i \le k \\ {\alpha}_i - 1
\;\;\text{if} \;\;\;i> k. \end{cases}\]
For every \(\beta \in I(\ell, m)\) and \(1 \le k \le \ell\), define \(\beta^{(k)} = (\beta_1, \dots, \beta_k) \in I(k, m)\).
For \(\gamma = (\gamma_1, \dots, \gamma_k) \in I(k, m)\) with \(\gamma \in \nabla ({\alpha}^{(k)})\), we define \[C_\gamma^{(k)} (\ell, V_m,
\mathcal{B}) :=\left\{L \in \Omega_\alpha (\ell, V_{m}, \mathcal{B}): \text{the first k pivots of} \;M_{\mathcal{B}}(L) \;\text{occur on the columns} \;\gamma_1, \dots, \gamma_k \right\}\]
Example 1. Suppose \({\alpha}= (3,4,6,7) \in I(4, 7)\). In this case, \(k =2\), \({\alpha}' = (3,4,5,6)\) and \(\widecheck{{\alpha}} = (3,4,6)\). On the other hand, if \({\alpha}= (3, 4, 6, 8)\), then we have \(k = 3\), \({\alpha}' = (3, 4, 6,
7)\) and \(\widecheck{{\alpha}} = (3,4,6)\).
Definition 3. Let \({\alpha}, \ell, m, {\alpha}', \widecheck{{\alpha}}\) be as in Definition 2 and as before, let \(\mathcal{B}= \{v_1, \dots, v_m\}\) be a fixed ordered basis of \(V_m\). We define
\(V_i = \text{span} \{v_1, \dots, v_i\}\) for \(i = 1, \dots, m\). In particular, we have the \(m-1\) dimensional subspace \(V_{m-1}\) of \(V_m\) that is spanned by \(\{v_1, \dots, v_{m-1}\}\).
\(\mathcal{B}'=\{v_1, \dots, v_{m-1}\}\).
In particular, when \(\ell \le m-1\), the Schubert subvarieties \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\) and \(\Omega_{\widecheck{\alpha}}
(\ell -1, V_{m-1}, \mathcal{B}')\) are well-defined.
Using the definition of \({\alpha}'\), we have \[\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}'):= \left\{L \in G(\ell, V_m) \middle\vert \begin{array}{l} \dim L \cap
V_{{\alpha}_i} \ge i \;\;\text{if} \;\;1 \le i \le k \\ \dim L \cap V_{{\alpha}_i - 1} \ge i \;\;\text{if} \;\; k < i \le \ell \end{array}\right\}.\]
Similarly, with the definition of \(\widecheck{{\alpha}}\), we have \[\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}'):= \{L \in G(\ell - 1, V_{m-1}) | \dim L \cap
V_{{\alpha}_i} \ge i \;\text{for all} \;1 \le i \le \ell -1\}.\]
It is evident that \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\subset \Omega_\alpha (\ell, V_{m}, \mathcal{B})\). Furthermore, \(\Omega_{\alpha'} (\ell, V_{m-1},
\mathcal{B}')\) comes up with its own cellular decomposition. In fact, we may write \[\label{a39} \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')= \bigsqcup_{\beta \in \nabla
({\alpha}')} C_\beta (\ell, V_{m - 1}, \mathcal{B}').\tag{11}\] Let us note an interesting coincidence below.
Lemma 2. For each \(\beta \in \nabla ({\alpha}')\), we have \(C_\beta (\ell, V_{m-1}, \mathcal{B}') = C_\beta (\ell, V_m, \mathcal{B})\).
Proof. Let \(\beta \in \nabla(\alpha')\). By definition, \[C_\beta(\ell, V_{m-1}, \mathcal{B}')
=
\left\{
L \in G(\ell, V_{m-1}) :
\dim(L \cap V_{\beta_i}) = i \text{ for all } i=1,\dots,\ell
\right\}.\] Since \(V_{m-1} \subset V_m\) and \(\mathcal{B}'=\{v_1,\dots,v_{m-1}\}\subset \mathcal{B}=\{v_1,\dots,v_m\}\), for any \(\beta\in\nabla(\alpha')\), the flag \[0 \subset V_{\beta_1} \subset \cdots \subset V_{\beta_\ell}\] is identical whether regarded as a flag in \(V_{m-1}\)
or in \(V_m\). The defining rank conditions of the Schubert cell \(C_\beta\) in \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\) and \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) coincide. Consequently, a subspace \(L \subset V_{m-1}\) satisfies the defining conditions of \(C_\beta(\ell, V_{m-1},
\mathcal{B}')\) if and only if it satisfies those of \(C_\beta(\ell, V_m, \mathcal{B})\). Hence, \[C_\beta(\ell, V_{m-1}, \mathcal{B}') = C_\beta(\ell, V_m, \mathcal{B}),\]
as required. ◻
Let us now introduce some more definitions.
Definition 4. Let \({\alpha}, \ell, k\) be as in Definition 2. Define a map \(\phi : \nabla
({\alpha}') \to \nabla ({\alpha})\) given by \[(\beta_1, \dots, \beta_k, \beta_{k+1}, \dots, \beta_\ell) \mapsto (\beta_1, \dots, \beta_k, \beta_{k+1}+1, \dots, \beta_\ell+1).\] Evidently, the map \(\phi\) is well-defined. The map \(\phi\) induces a map \[\Phi : \{C_\beta (\ell, V_{m-1}, \mathcal{B}') | \beta \in \nabla ({\alpha}')\} \to \{C_\beta (\ell,
V_{m}, \mathcal{B}) | \beta \in \nabla ({\alpha})\}\] given by \[\label{Phi}
\Phi(C_\beta (\ell, V_{m-1}, \mathcal{B}')) = C_{\phi(\beta)} (\ell, V_m, \mathcal{B}).\qquad{(2)}\] The Schubert cells of \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) that are not images of \(\Phi\) are called the missing cells.
Indeed, the definition of missing cells is not vacuous. One can easily check that the cell corresponding to \(\beta = (1, \dots, \ell)\) is a missing cell. Moreover, there are some missing cells of very high dimensions.
For example, the cell corresponding to \({\alpha}'' = ({\alpha}_1, \dots, {\alpha}_k, {\alpha}_{k} + 1, {\alpha}_{k+2}, \dots, {\alpha}_{\ell})\) is a missing cell with dimension \(\delta
({\alpha}'') = \delta ({\alpha}) - ({\alpha}_{k+1} - {\alpha}_{k} - 1)\). In fact, one could easily check that any \(\gamma \in I(\ell, m)\) with \(\gamma_{k+1} - \gamma_k =
1\) corresponds to a missing cell.
Lemma 3. Let \({\alpha}, \ell, k\) be as in Definition 2. Define \({\alpha}_0 = (1, \dots, \ell - 1,
m)\). For \(\beta \in \nabla ({\alpha}')\), we have
The first assertion follows since the left hand side equals \(q^{\delta(\phi(\beta))}\), while the right hand side equals \(q^{\delta (\beta) + \ell - k}\) and by definition \(\delta (\phi(\beta)) = \delta(\beta) + \ell -k\).
Let us denote by \(I_1 = \{\gamma \in \nabla ({\alpha}) : \gamma = \phi (\beta) \;\text{for some} \;\beta \in \nabla ({\alpha}')\}\) and \(I_2 = \nabla ({\alpha}) \setminus I_1\).
Note that for every \(\gamma \in I_2\), the cell \(C_\gamma (\ell, V_m, \mathcal{B})\) is a missing cell, as defined above. We have \[\begin{align} &
\;\;|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\ell - k} |\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| \\ &= \sum_{\gamma \in \nabla ({\alpha})} |C_{\gamma} (\ell, V_m, \mathcal{B})| - q^{\ell - k} \sum_{\beta \in \nabla
({\alpha}')} |C_\beta(\ell, V_{m-1}, \mathcal{B}')| \\ &=\sum_{\beta \in \nabla ({\alpha}')} (|C_{\phi (\beta)} (\ell, V_m, \mathcal{B}))| - q^{\ell - k} |C_\beta(\ell, V_{m-1}, \mathcal{B}')|) + \sum_{\gamma \in I_2} |C_{\gamma} (\ell,
V_m, \mathcal{B})| \\ &= \sum_{\gamma \in I_2} |C_{\gamma} (\ell, V_m, \mathcal{B})|.
\end{align}\] It is easy to see that for any \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell)\), with \(k \le \ell -1\), the cell corresponding to the tuple \(\tilde{\alpha}= ({\alpha}_1, \dots, {\alpha}_k, {\alpha}_k + 1, {\alpha}_{k+2}, \dots, {\alpha}_\ell)\) is a missing cell. Moreover, \[q^{{\alpha}_{k+1} - {\alpha}_k - 1} q^{\delta (\tilde{\alpha})} =
q^{\delta ({\alpha})}.\] In particular, if \(k = \ell - 1\), then \(\tilde{\alpha}= ({\alpha}_1, \dots, {\alpha}_{\ell - 1}, {\alpha}_{\ell - 1} + 1)\) and the same equality as above
holds. This proves that \[q^{{\alpha}_{k+1} - {\alpha}_k - 1} (|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\ell - k}|\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|) \ge q^{\delta ({\alpha})}\] for every \({\alpha}\in I(\ell, m)\). Moreover, if \({\alpha}\neq {\alpha}_0\), then we can produce another missing cell \(\bar{{\alpha}}\) such that \({\alpha}_{k+1} - {\alpha}_k - 1 + \delta (\bar{\alpha}) \ge \delta ({\alpha}')\) as given below. Let \[k' = \begin{cases} 1 \;&\text{if} \;{\alpha}_1 > 1 \\ \min\{j : {\alpha}_j -
{\alpha}_{j-1} > 1 \} \;&\text{if} \;{\alpha}_1 = 1 \end{cases}.\] Clearly, \(k'\) is well-defined, thanks to the condition that \({\alpha}\neq {\alpha}_0\). Define \(\bar{\alpha}= (\bar{\alpha}_1, \dots, \bar{\alpha}_\ell)\) as follows: \[\bar{\alpha}_i = \begin{cases} {\alpha}_i - 1 &\text{if} \;i = k' \\ {\alpha}_{k} + 1 \;&\text{if} \;i = k + 1 \\
{\alpha}_i \;&\text{otherwise} \end{cases}\] One checks that \({\alpha}_{k+1} - {\alpha}_k - 1 + \delta(\bar{\alpha}) = \delta ({\alpha}) - 1 \ge \delta ({\alpha}')\).
This completes the proof. ◻
We state and prove the following bound. Undoubtedly, the bound looks unmotivated at this stage, but it will be central in the proof of our main theorem.
Suppose that \({\alpha}= (1, \dots, \ell - 1, m)\). From Lemma 3, we have \[q^{{\alpha}_{k+1} -{\alpha}_k - 1}
(|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\ell - k}|\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|) \ge q^{\delta ({\alpha})} = q^{m - \ell}.\] On the other hand, we have \(|\Omega_{\widecheck{\alpha}} (\ell
-1, V_{m-1}, \mathcal{B}')| = 1.\) Combining all these, we see that the right-hand side of the desired inequality equals \(q^{\delta ({\alpha}')}\). The assertion follows since \(q^{\delta ({\alpha})} > q^{\delta ({\alpha}')}\).
Suppose that \({\alpha}\) is not of the form \((1, \dots, \ell - 1, m)\). Again Lemma 3 shows that the
left-hand side of the desired inequality is bigger than or equal to \(q^{\delta({\alpha})} + q^{\delta ({\alpha}')}\). On the other hand Proposition 6
shows that, \[\begin{align} q^{m - \ell} |\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')| + q^{\delta ({\alpha}')} - q^{\delta ({\alpha})} &< \left(\frac{q}{q-1}\right)^{\ell - 1} q^{\delta
(\widecheck{{\alpha}})}q^{m-\ell} + q^{\delta ({\alpha}')} - q^{\delta ({\alpha})}. \\ &=\left(\frac{q}{q-1}\right)^{\ell - 1}q^{\delta ({\alpha})} + q^{\delta ({\alpha}')} - q^{\delta ({\alpha})}.
\end{align}\] In order to prove the assertion, it is enough to show that \[q^{\delta ({\alpha})} + q^{\delta ({\alpha}')} > \left(\frac{q}{q-1}\right)^{\ell - 1} q^{\delta ({\alpha})} + q^{\delta ({\alpha}')} -
q^{\delta ({\alpha})},\] which is equivalent to the inequality \[\left(\frac{q}{q-1}\right)^{\ell - 1} < 2.\] The above inequality holds if and only if \[\left(\frac{q}{q-1}\right) <
2^{\frac{1}{\ell - 1}} \iff \frac{1}{q-1} < 2^{\frac{1}{\ell - 1}} - 1\iff q > 1 + \frac{1}{2^{\frac{1}{\ell - 1}} - 1} = \frac{2^{\frac{1}{\ell - 1}}}{2^{\frac{1}{\ell - 1}} - 1} = q_0 (\ell).\]
This completes the proof. ◻
Before concluding the subsection, we remark that the function \(q_0 (\ell)\) is increasing in \(\ell\).
3.3 A decomposition of Schubert subvarieties and hyperplane sections↩︎
This subsection is motivated by [15]. We will continue with the notations introduced in the previous sections and subsections. We note that the Schubert subvariety
\(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\) is naturally contained in the Schubert subvariety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). One can prove it directly using the
definitions. On the other hand, this is a direct consequence of Lemma 2. Independently, one can also see that \(\Omega_{\alpha'} (\ell, V_{m-1},
\mathcal{B}')= \Omega_\alpha (\ell, V_{m}, \mathcal{B})\cap G(\ell, V_{m-1}) \subseteq \Omega_\alpha (\ell, V_{m}, \mathcal{B})\). Lemma 2 implies that
\[\label{st1}
\Omega_\alpha (\ell, V_{m}, \mathcal{B})= \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\bigsqcup \left(\bigsqcup_{\beta \in \Delta ({\alpha}') \cap \nabla({\alpha})} C_\beta (\ell, V_m, \mathcal{B}) \right).\tag{12}\] As in [15], we decompose \(\displaystyle\left(\bigsqcup_{\beta \in \Delta ({\alpha}') \cap \nabla({\alpha})} C_\beta (\ell, V_m, \mathcal{B})
\right)\) into strings to facilitate computations. First, we note that \[\label{pivot} \Delta ({\alpha}') \cap \nabla({\alpha}) = \{\beta \in \nabla ({\alpha}) : \beta_\ell =
m\}.\tag{13}\] We recall that the condition \(\beta_\ell = m\) mentioned above is a consequence of our running assumption that in our Schubert variety \(\Omega_\alpha (\ell, V_{m},
\mathcal{B})\), the index \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell)\) satisfies \({\alpha}_\ell = m\). Before delving into the technical details, for the sake of clarity, let us
revisit our notations related to matrix representations from 4 . Recall that the bijection in 4 associates every element of \(G(\ell, V_m)\) to a matrix in \(\mathcal{M}(\ell, m)\) with respect to the fixed basis \(\mathcal{B}= \{v_1, \dots, v_m\}\). Let us denote by \(\mathcal{M}_{{\alpha}} (\ell, m)\) the image of
this map restricted to \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) and \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) that of \(\Omega_{\alpha'} (\ell, V_{m-1},
\mathcal{B}')\). In particular, we have the following description of \(\mathcal{M}_{\alpha}(\ell, m)\)\[\mathcal{M}_{\alpha}(\ell, m) = \{M \in \mathcal{M}(\ell, m) | \;\text{the pivot
in the i-th row occur on \beta_ith column with} \;\beta_i \le {\alpha}_i, \;\forall i\}.\] The description of \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) can be given similarly. An important remark on \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) is in order. As per our continued assumptions, we are treating the elements of \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) as \(\ell \times (m-1)\) matrices. However, when we say that we view \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) as a subset of \(\mathcal{M}_{\alpha}(\ell, m)\), we
use the natural inclusion \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) into \(\mathcal{M}_{\alpha}(\ell, m)\) by juxtaposing a zero column to the right of each element of \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\). Thus, we may regard the elements of \(\mathcal{M}_{{\alpha}'}(\ell, m-1)\) as \(\ell \times m\) matrices with
entries in \({\mathbb{F}}\) so that they can safely be regarded as elements of \(\mathcal{M}_{\alpha}(\ell, m)\). Inspired from [15], we will provide a decomposition of \(\mathcal{M}_{\alpha}(\ell, m)\) into \(\mathcal{M}_{{\alpha}'} (\ell, m-1)\) and union of strings.
This decomposition is essentially obtained through 12 above. Let us write \[T_{\alpha}(\ell, m) := \{M \in \mathcal{M}_{\alpha}(\ell, m) | \text{the pivot of the last row of \mathcal{M} occurs on m-th
column}\}.\] One can see that, under the correspondence in 4 , the set \(T_{\alpha}(\ell, m)\) corresponds to \[\mathfrak{F}_{\alpha}(\ell, V_m, \mathcal{B}) :=
\Omega_\alpha (\ell, V_{m}, \mathcal{B})\setminus \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')= \bigsqcup_{\beta \in \Delta ({\alpha}') \cap \nabla ({\alpha})} C_\beta (\ell, V_m, \mathcal{B}).\] Indeed, the second equality is a
consequence of 12 . It is understood that a stratification of \(T_{\alpha}(\ell, m)\) into strings gives rise to a decomposition of \(\mathfrak{F}_{\alpha}(\ell, V_m,
\mathcal{B})\) into smaller components.
Let \(M \in T_{{\alpha}}(\ell, m)\). We may write \[\label{cm} M = \begin{pmatrix} \widecheck{M} && {\boldsymbol{0}} \\ c_M && 1
\end{pmatrix},\tag{14}\] where \(\widecheck{M} \in \mathcal{M}_{\widecheck{{\alpha}}}(\ell - 1, m-1)\), the symbol \({\boldsymbol{0}}\) denotes a column matrix of length \((\ell - 1)\) with all entries equal to \({0}\), whereas \(c_M\) is a row vector of length \(m-1\), and \(1\) is the multiplicative identity of \({\mathbb{F}}\). Since \(M\) is in right-row-reduced echelon form, so is \(\widecheck{M}\), which justifies the inclusion \(\widecheck{M} \in \mathcal{M}_{\widecheck{{\alpha}}}(\ell - 1, m-1)\). The pivot columns of \(\widecheck{M}\)
are determined by those of \(M\), and the entries in \(c_M\) on the pivot columns of \(\widecheck{M}\) are zero. Let us denote \((M_{n_1}, \dots, M_{n_{m-\ell}})\) the entries in the non-pivotal columns of \(c_M\), where \(n_1 < \cdots < n_{m-\ell}\). This defines a map \[s : T_{\alpha}(\ell, m) \to {\mathbb{F}}^{m-\ell}.\] The map \(s\) is clearly surjective. We claim that \(s^{-1} (\nu)\) can be identified with \(\mathcal{M}_{\widecheck{{\alpha}}} (\ell- 1, m-1)\) for any \(\nu = (\nu_1, \dots, \nu_{m-\ell}) \in {\mathbb{F}}^{m-\ell}\). Define \[\label{bij2}
\phi_\nu : \mathcal{M}_{\widecheck{{\alpha}}} (\ell-1, m-1) \to s^{-1} (\nu) \;\;\;\text{by} \;\;\;M' \mapsto \begin{pmatrix} {M'} && 0 \\ c_{M', \nu} && 1 \end{pmatrix},\tag{15}\] where \(c_{M', \nu}\) is uniquely determined by the non-pivot positions of \(M'\) and \(\nu\). Clearly, the map \(\phi_\nu\) is
a bijection. Using the bijection \(\mathcal{M}_{\widecheck{{\alpha}}} (\ell - 1, m-1) \longleftrightarrow \Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\), we note that \(|s^{-1}
(\nu)| = |\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')|\) for all \(\nu \in {\mathbb{F}}_q^{m-\ell}\). Moreover, it follows trivially that \(s^{-1} (\nu) \cap s^{-1}
(\nu') = \emptyset\) whenever \(\nu \neq \nu'\). Thus, \[\label{t} \mathcal{M}_{\alpha}(\ell, m) = \mathcal{M}_{{\alpha}'} (\ell, m-1) \bigsqcup
T_{\alpha}(\ell, m) \;\;\;\text{where} \;\;\; T_{\alpha}(\ell, m) = \bigsqcup_{\nu \in {\mathbb{F}}^{m-\ell}} s^{-1} (\nu).\tag{16}\] We call \(s^{-1} (\nu)\), the \(\nu\)-th string of \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) with respect to \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\). We look at a
quick enumerative application of the above decomposition.
Proposition 9. Let \({\mathbb{F}}= {\mathbb{F}}_q\). Then
with \(k\) as defined above and \(q > q_0 (\ell)\), we have \[\begin{align} q^{{\alpha}_{k+1} -{\alpha}_k - 1} (|\Omega_\alpha (\ell, V_{m},
\mathcal{B})| - & q^{\ell - k}|\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|) \\ &> q^{\delta ({\alpha}')} + (|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - |\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|) - q^{\delta
({\alpha})}.
\end{align}\]
Proof. Part (a) follows at once from 16 and the discussion above, while part (b) is a consequence of part (a) and Theorem 8. ◻
We would like to understand how certain hyperplanes behave with the above decomposition of \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). Recall the following inclusion of \(\Omega_\alpha (\ell,
V_{m}, \mathcal{B})\;\text{and} \; \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\) in \(G(\ell, V_m)\).
Let \(\Pi_{\alpha}\) be a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) given by a homogeneous polynomial \(F_{\alpha}(X_\beta | \beta \in \nabla
({\alpha}))\).
Note that \(\Pi_{\alpha}\) does not contain \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). This follows from the non-degeneracy of \(\Omega_\alpha
(\ell, V_{m}, \mathcal{B})\) in \({\mathbb{P}}^{k_{\alpha}- 1}\). Thus \(F_{\alpha}\neq 0\). We write \[F_{\alpha}= F_{{\alpha}'} (X_\beta| \beta \in
\nabla ({\alpha}') )+ F' (X_\beta | \beta \in \nabla ({\alpha}) \cap \Delta ({\alpha}')).\] Then \(\Pi_{\alpha}\) restricts to a hyperplane \(\Pi_{{\alpha}'} =
\Pi_{\alpha}\cap {\mathbb{P}}^{k_{{\alpha}'} - 1}\) of \({\mathbb{P}}^{k_{{\alpha}'} - 1}\) given by the vanishing of \(F_{{\alpha}'}\) in \({\mathbb{P}}^{k_{{\alpha}'} - 1}\), provided \(F_{{\alpha}'} \neq 0\).
Suppose that \(\Pi_{\alpha}\) contains \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\). The non-degeneracy of \(\Omega_{\alpha'}
(\ell, V_{m-1}, \mathcal{B}')\) in \({\mathbb{P}}^{k_{{\alpha}'} - 1}\) implies that the assumption is equivalent to the statement that \(F_{{\alpha}'} = 0\).
Consequently, the polynomial \(F_{\alpha}\) is a linear combination of \(X_\beta\) as \(\beta\) varies in the set \(\nabla
({\alpha}) \cap \Delta({\alpha}')\). We note that for each such \(\beta \in \nabla ({\alpha}) \cap \Delta({\alpha}')\), we have \(\beta_\ell = {\alpha}_\ell = m\). It also
follows from 16 that \[\label{t1} |\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Pi_{\alpha}\cap \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| +
\sum_{\nu \in {\mathbb{F}}_q^{m - \ell}} |\Pi_{\alpha}\cap s^{-1} (\nu)|.\tag{17}\] Since \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\subseteq \Pi_{\alpha}\), we see that \(|\Pi_{\alpha}\cap \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| = |\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|\). Consequently, the equality above translates to \[\label{t1465} |\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| + \sum_{\nu \in {\mathbb{F}}_q^{m - \ell}} |\Pi_{\alpha}\cap s^{-1} (\nu)|.\tag{18}\]
We will, until the end of this section, assume that \(\Pi_{\alpha}\) is a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) satisfying (II) above. In order to compute \(|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})|\), it is now imperative to better understand the set \(\Pi_{\alpha}\cap s^{-1} (\nu)\) for each \(\nu \in
{\mathbb{F}}_q^{m-\ell}\). For a fixed \(\nu \in {\mathbb{F}}_q^{m - \ell}\), recall that set \(s^{-1} (\nu)\) is given by \[\left \{\begin{pmatrix}
\widecheck{M} && \boldsymbol{0} \\ c && 1\end{pmatrix} : \widecheck{M} \in \mathcal{M}_{\widecheck{{\alpha}}} (\ell - 1, m-1) \right\},\] which as per the above discussion is in one-one correspondence with \(M_{\widecheck{{\alpha}}} (\ell - 1, m-1)\). We thus obtain the following bijections given by natural identifications: \[\label{bijn} s^{-1} (\nu) \longleftrightarrow
\mathcal{M}_{\widecheck{{\alpha}}} (\ell - 1, m-1) \longleftrightarrow \Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}').\tag{19}\]
Take \(M \in s^{-1} (\nu)\). For \(\beta \in \Delta ({\alpha}') \cap \nabla ({\alpha})\), we see that \(\beta_\ell = m\) and consequently, \(X_{\beta} (M) = X_{\widecheck{\beta}, \nu} (\widecheck{M})\), where the latter is determined by taking the \((\ell - 1) \times (\ell - 1)\) minor of \(M\) given
by the columns numbered \(\beta_1, \dots, \beta_{\ell - 1}\) and the first \(\ell - 1\) rows. For \[F_{\alpha}= \displaystyle\sum_{\beta \in \Delta ({\alpha}')
\cap \nabla ({\alpha})} c_\beta X_\beta, \;\;\text{we write} \;\;F_{\widecheck{{\alpha}}, \nu} = \displaystyle\sum_{\beta \in \Delta ({\alpha}') \cap \nabla ({\alpha})}c_\beta X_{\widecheck{\beta}, \nu}.\] For each \(M
\in \Omega_\alpha (\ell, V_{m}, \mathcal{B})\setminus \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\), we have \(F_{\alpha}(M) = F_{\widecheck{{\alpha}}, \nu} (M)\). Before proceeding further, we remark that
the evaluation is independent of \(\nu\). Namely, if \(M, M' \in \Omega_\alpha (\ell, V_{m}, \mathcal{B})\setminus \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\), with
\(M \in s^{-1} (\nu)\) and \(M' \in s^{-1} (\mu)\) for some distinct \(\nu, \mu \in {\mathbb{F}}_q^{m-\ell}\), but \(\widecheck{M} = \widecheck{M'}\), then \(F_{{\alpha}} (M) = F_{{\alpha}} (M')\). In particular, we have \[|\{M \in s^{-1} (\nu) : F_{{\alpha}} (M) = 0\}| =
|\{M \in s^{-1} (\mu) : F_{{\alpha}} (M) = 0\}| \;\;\text{for} \;\;\nu, \mu \in {\mathbb{F}}_q^{m-\ell}.\] Combining this with 18 , we obtain \[\label{t2}
|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| + q^{m-\ell} |\Pi_{\alpha}\cap s^{-1} (\nu)|\tag{20}\]
Thus we may fix \(\nu \in {\mathbb{F}}_q^{m-\ell}\) and focus on computing \(|\Pi_{\alpha}\cap s^{-1} (\nu)|\). We have already identified \(s^{-1}
(\nu)\) with the Schubert subvariety \(\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\). Let us ponder on the Schubert subvariety \(\Omega_{\widecheck{\alpha}} (\ell -1,
V_{m-1}, \mathcal{B}')\) a little more before proceeding further. Indeed, \(\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\subset G(\ell - 1, V_{m-1})\). Now \(G(\ell -
1, V_{m-1})\) is embedded inside the projective space \({\mathbb{P}}\left( \displaystyle{\bigwedge^{\ell-1}} V_{m-1}\right)\) which is a projective space of dimension \(\widecheck{k} -
1\), where \(\widecheck{k} = {m - 1 \choose \ell - 1}\). Moreover, as usual, the Schubert subvariety \(\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\) is obtained
by taking an appropriate linear section of \(G(\ell - 1, V_{m-1})\) given by the polynomials \(X_{\widecheck{\beta}}\) where \(\widecheck{\beta} \in I(\ell - 1,
m-1)\) with \(\widecheck{\beta} \not\leq \widecheck{{\alpha}}\). We see that these \(\widecheck{\beta} = (\widecheck{\beta}_1, \dots, \widecheck{\beta}_{\ell - 1})\)-s get naturally
identified with \(\beta \in \Delta ({\alpha}) \subset I(\ell, m)\) with \(\beta_i = \widecheck{\beta}_i\) for \(i = 1, \dots, \ell - 1\) and \(\beta_{\ell} = m\). In fact, if we look at the identification \(X_{\widecheck{\beta}}\) as an element of \(\bigwedge^{m - \ell} V_{m-1} \subset \bigwedge^{m-\ell}
V_m\), then it is actually equal to \(X_{\beta}\). At any rate, we may assume that \(\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\) is embedded in \({\mathbb{P}}^{k_{\widecheck{{\alpha}} - 1}}\), where the latter is given by the linear subspace of \({\mathbb{P}}^{k_{\widecheck{{\alpha}}} - 1}\) mentioned above. Thus, under the identification
of \(s^{-1} (\nu)\) with \(\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\) described in 19 , we see that the set \(s^{-1}
(\nu) \cap \Pi_{\alpha}\) gets identified with the set \(\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\cap \Pi_{\widecheck{{\alpha}}}\) where \(\Pi_{\widecheck{{\alpha}}}\) is the hyperplane of \({\mathbb{P}}^{k_{\widecheck{{\alpha}}} - 1}\).
To summarize, let \(\Pi_{\alpha}\) is a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) satisfying (I) and (II) above. Then \(\Pi_{\alpha}\) is given by a
linear homogeneous polynomial \[F_{\alpha}= \sum_{{\beta \in \Delta ({\alpha}') \cap \nabla ({\alpha}) }} c_\beta X_\beta.\] Then \(\Pi_{\alpha}\) restricts to the hyperplane \(\Pi_{\widecheck{{\alpha}}}\) given by \[F_{\widecheck{{\alpha}}} = \sum_{{\beta \in \Delta ({\alpha}') \cap \nabla ({\alpha})}} c_\beta X_{\widecheck{\beta}}.\]
Remark 10. We readily see that a hyperplane satisfying (I) and (II) above can not contain \(s^{-1}(\nu)\). Otherwise, it follows from 19 , 20 , and
Proposition 9 that \(|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Omega_\alpha (\ell, V_{m}, \mathcal{B})|\), which implies that
\(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\subset \Pi_{\alpha}\), contradicting (I). In particular, we have \(|\Pi_{\alpha}\cap s^{-1} (\nu)| < |s^{-1} (\nu)|.\)
Let us now introduce some notations for ease of reference.
Definition 5. Let \({\alpha}, {\alpha}',\widecheck{{\alpha}}, \ell, m\) be as defined above. With \({\mathbb{F}}= {\mathbb{F}}_q\), we define
\(e_{\alpha}(\ell, m) := \max \{|\Omega_\alpha (\ell, V_{m}, \mathcal{B})\cap \Pi_{\alpha}| : \Pi_{\alpha}\;\text{is a hyperplane of} \;{\mathbb{P}}^{k_{{\alpha}} - 1}\}\).
We conclude this section with the following Proposition summarizing the discussion in the current section which can be directly used in the proof of our main theorem.
Proposition 11. Let \({\mathbb{F}}= {\mathbb{F}}_q\). If \(\Pi_{\alpha}\) is a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) such
that \(\Pi_{\alpha}\) contains \(\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')\), then \[|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| \le
|\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| + q^{m - \ell} e_{\widecheck{{\alpha}}}(\ell - 1, m-1).\]
Proof. From 20 , we know that \(|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Pi_{\alpha}\cap \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')| + q^{m-\ell}
|\Pi_{\alpha}\cap s^{-1} (\nu)|\). Moreover, as mentioned in Remark 10, the hyperplane \(\Pi_{\alpha}\) does not contain \(s^{-1} (\nu)\). Thus \(|\Pi_{\alpha}\cap s^{-1} (\nu)| = |\Pi_{\widecheck{{\alpha}}} \cap \Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')| \le e_{\widecheck{{\alpha}}}(\ell - 1,
m-1).\) The assertion now follows trivially. ◻
Throughout this section, we will assume that \({\mathbb{F}}_q\) is a finite field with \(q\) elements. All the notations \(V_m, {\alpha}, {\alpha}',
\widecheck{{\alpha}}, \Omega_\alpha (\ell, V_{m}, \mathcal{B}), \Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}'), \;\text{and} \;\Omega_{\widecheck{\alpha}} (\ell -1, V_{m-1}, \mathcal{B}')\) are as discussed in the previous sections.
Before presenting our first result, which is a Schubert-analog of [15], we define a special type of containment of Schubert subvarieties.
Definition 6. Let \(\ell \le m - 1\). Let \(\mathcal{B}\) be as above. Let \(W_{m-1}\) be an \((m-1)\) dimensional subspace of \(V_m\) with an ordered basis \(\mathcal{B}_1 = \{w_1, \dots, w_{m-1}\}\). As usual, we denote by \(W_i
= Span \{w_1, \dots, w_i\}\) for \(i =1, \dots, m-1\). For \({\alpha}' = ({\alpha}_1, \dots, {\alpha}_k, {\alpha}_{k+1} - 1, \dots, {\alpha}_\ell - 1)\) as defined above, we say
that \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\) is strictly contained in* \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) if \(V_{{\alpha}_i} =
W_{{\alpha}_i} \;\text{for} \;i = 1, \dots, k\) and \(V_{{\alpha}_k} \subset W_{m-1} \subset V_m\). In this case, we write \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le
\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) and define \[\mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B}) := \{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le \Omega_\alpha (\ell, V_{m}, \mathcal{B})\}.\]*
Remark 12. Before proceeding, it is worth understanding the above definition and its consequences a bit more carefully. Let us take \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \in
\mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})\). We write \(\mathcal{B}_1 = \{w_1, \dots, w_{m-1}\}\) which serves as the fixed ordered basis of \(W_{m-1}\). Let us extend it to an
ordered basis \(\tilde{B} = \{w_1, \dots, w_{m-1}, w_m\}\) of \(V_m\). Let us consider the Schubert subvariety \(\Omega_{{\alpha}} (\ell, V_m,
\tilde{\mathcal{B}})\). This is given by the flag \[V_{{\alpha}_1} \subset \cdots \subset V_{{\alpha}_k} \subset W_{m-(\ell - k) + 1} \subset \cdots \subset W_{m-1} \subset V_m.\] Note that, for any \(L \in \Omega_\alpha (\ell, V_{m}, \mathcal{B})\), we have the Schubert conditions: \(\dim (L \cap V_{{\alpha}_i}) \ge i\) for \(i = 1, \dots, \ell\). It is clear
that \(i = m - (\ell-k) + 1, \dots, m-1\), we have \(\dim (L \cap W_i) \ge i\). As a consequence, we have \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\subset
\Omega_{{\alpha}} (\ell, V_m, \tilde{\mathcal{B}})\). A similar argument establishes the reverse inclusion, consequently implying that \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})= \Omega_{{\alpha}} (\ell, V_m,
\tilde{\mathcal{B}})\). Moreover, as per Definition 3 (b), we have \(\tilde{\mathcal{B}}'= \mathcal{B}_1\).
If a hyperplane \(\Pi_{\alpha}\) of \({\mathbb{P}}^{k_{\alpha}- 1}\) contains \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\) for some
\(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \in \mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})\), then we may assume, after a change of coordinates (induced by the change of bases from \(\mathcal{B}\) to \(\tilde{\mathcal{B}}\)) that \(\Pi_{\alpha}\) contains \(\Omega_{\alpha'} (\ell, V_{m-1},
\mathcal{B}')\). By doing this, we can use the tools we have developed above, as in Proposition 11, among others.
Next, suppose \(\Pi_{\alpha}\) is a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\) such that \(\Pi_{\alpha}\) does not contain \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\). Again, with respect to the new coordinate system on \({\mathbb{P}}^{k -1}\) and in particular \({\mathbb{P}}^{k_{\alpha}- 1}\), we may assume that \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) = {\mathbb{P}}^{k_{{\alpha}'} - 1} \cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})\).
We see that \(\Pi_{\alpha}\cap {\mathbb{P}}^{k_{{\alpha}'} - 1}\) is a hyperplane of \({\mathbb{P}}^{k_{{\alpha}'} - 1}\) and we have \(|\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \cap \Pi_{\alpha}| \le e_{{\alpha}'}(\ell, m-1)\).
We begin with the following proposition concerning the cardinalities of \(\mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})\) and some of its special subsets.
Proposition 13. With the notations as above, we have \[\label{eq} |\mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})| = \frac{q^{m - {\alpha}_k} - 1}{q-1}.\qquad{(3)}\] Moreover,
for \(L \in \Omega_\alpha (\ell, V_{m}, \mathcal{B})\), we have \[\label{ineq}|\{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \in \mathfrak{S}_{\alpha}(\ell, V_m,
\mathcal{B}) : L \in \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1\}| \ge \frac{q^{(m - {\alpha}_k) - (\ell - k)} - 1}{q-1}.\qquad{(4)}\]
Proof. Note that \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le \Omega_\alpha (\ell, V_{m}, \mathcal{B})\) if and only if \(V_{{\alpha}_i} = W_{{\alpha}_i}\) for all
\(i= 1, \dots, k\) and \(V_{{\alpha}_k} \subset W_{m-1} \subset V_m\). Moreover, for every pairwise distinct \(W_{m-1}\) and \(W_{m-1}'\) satisfying the above condition, we get pairwise distinct Schubert subvarieties \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\) and \(\Omega_{{\alpha}'} (\ell, W_{m-1}', \mathcal{B}_1')\), where \(\mathcal{B}_1'\) is obtained by using the partial flag obtained from the partial flag defining \(\mathcal{B}_1\) by replacing \(W_{m-1}\) with \(W_{m-1}'\). Thus \[|\mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})| = |\{W_{m-1} :
V_{{\alpha}_k} \subset W_{m-1} \subset V_m\}| = \frac{q^{m - {\alpha}_k} - 1}{q-1}.\] This proves ?? . Using a similar argument, we see that \[\begin{align} &|\{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) : L
\in \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le \Omega_\alpha (\ell, V_{m}, \mathcal{B})\}| \\ &=|\{W_{m-1} : L + V_{{\alpha}_k} \subset W_{m-1} \subset V_m\}| \\ &=|\{W_{m-1} : L + V_{{\alpha}_k} \subset W_{m-1} \subset V_m, \;\dim
(L + V_{{\alpha}_k}) \le {\alpha}_k + \ell - k\}| \\ &\ge |{\mathbb{P}}^{m - ({\alpha}_k + \ell - k)-1}| = \frac{q^{(m - {\alpha}_k) - (\ell - k)} - 1}{q-1}.
\end{align}\] This completes the proof. ◻
Lemma 4. Let \(\Pi_{\alpha}\) be a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\). Then \[|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m},
\mathcal{B})| \le \left(\frac{q^{m-{\alpha}_k} - 1}{q^{(m - {\alpha}_k) - (\ell - k)} - 1}\right) M_{\alpha},\] where \(M_{\alpha}= \max\{|\Pi_{\alpha}\cap \Omega_{{\alpha}'} (\ell, V_{m-1}, B_1)| :
\Omega_{{\alpha}'} (\ell, V_{m-1}, B_1) \in \mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})\}.\)
Proof. We define the following incidence set \(\mathfrak{I}\) and count it in two ways: \[\mathfrak{I} := \left\{\left(L, \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \right)
\in \Omega_\alpha (\ell, V_{m}, \mathcal{B})\times \mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B}) : \;L \in \Pi_{\alpha}\cap \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \right\}.\] One one hand, we have \[\begin{align} |\mathfrak{I}| &= \sum_{L \in \Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})} \#\{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) | L \in \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le
\Omega_\alpha (\ell, V_{m}, \mathcal{B})\} \\
&\ge \sum_{L \in \Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})} \frac{q^{(m - {\alpha}_k) - (\ell - k)} - 1}{q-1} \\
&= |\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| \frac{q^{(m - {\alpha}_k) - (\ell - k)} - 1}{q-1}.
\end{align}\] The inequality above is a consequence of ?? . On the other hand, we have \[\begin{align} |\mathfrak{I}| &= \sum_{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\le \Omega_\alpha (\ell, V_{m},
\mathcal{B})} \#\{L \in \Pi_{\alpha}\cap \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\} \\
&\le \sum_{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le \Omega_\alpha (\ell, V_{m}, \mathcal{B})} M_{\alpha}
=|\{\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \le \Omega_\alpha (\ell, V_{m}, \mathcal{B})\}| M_{\alpha}= \frac{q^{m - {\alpha}_k} - 1}{q-1} M_{\alpha}.
\end{align}\] The last equality follows from ?? . We have thus obtained \[|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| \frac{q^{(m - {\alpha}_k) - (\ell - k)} - 1}{q-1} \le |\mathfrak{J}| \le \frac{q^{m -
{\alpha}_k} - 1}{q-1} M_{\alpha}.\] The assertion now follows trivially. ◻
Let us now include an auxiliary Lemma that will be useful in the computations.
Proof. Note that the inequality in the assertion is equivalent to the inequality: \[\begin{align} q^{(m - {\alpha}_k) - (\ell - k)} (|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - & q^{\ell - k}
|\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|)\\ & > (|\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - |\Omega_{\alpha'} (\ell, V_{m-1}, \mathcal{B}')|) - q^{\delta ({\alpha})} + q^{\delta({\alpha}')} .
\end{align}\] It follows from the definition of \(k\) that \(m = {\alpha}_\ell = {\alpha}_{k+1}+ (\ell - k - 1)\). Consequently, \((m - {\alpha}_k) - (l - k)
= {\alpha}_{k+1} - {\alpha}_k - 1\). The assertion now follows from Proposition 9 (b). ◻
Remark 14. Before proving our main theorem, let us warm up by considering a few trivial cases. As before, let \(1 \le \ell < m\), \(V_m\) a vector space of
dimension \(m\) over \({\mathbb{F}}_q\) with an ordered basis \(\mathcal{B}\) and \({\alpha}\in I(\ell, m)\). Our goal will
be to prove the following results, when \(q > q_0 (\ell)\):
\(e_{\alpha}(\ell, m) = |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\delta({\alpha})}\).
If \(\Pi_{\alpha}\) is a hyperplane such that \(|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\delta({\alpha})}\),
then \(\Pi_{\alpha}\) is Schubert decomposable.
As we have mentioned in Remark 5 (a), it was proved by Ghorpade and Singh [7] that if
\(\Pi_{\alpha}\) is a hyperplane that satisfies \(|\Pi_{\alpha}\cap \Omega_\alpha (\ell, V_{m}, \mathcal{B})| = |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\delta({\alpha})}\) and \(\Pi_{\alpha}\) is decomposable, that is \(\Pi_{\alpha}\) is given by a decomposable hyperplane of \({\mathbb{P}}^{k-1}\), then \(\Pi_{\alpha}\) is Schubert decomposable. Thus, while proving our main theorem, we will only restrict ourselves to proving that a hyperplane attaining the above bound is decomposable. Also, we will settle a few extreme cases
completely here, so that we can ignore them in our main proof.
Suppose \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell)\) are consecutive. In this case, \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\) is the Grassmannian \(G(\ell, V_m)\) and the result is known in this case due to Nogin [8]. Moreover, a method similar to this article was used in [15] to obtain an alternative proof of this result.
Let us now consider the case when \(\ell = m - 1\). In this case \(G(\ell, V_{m})\) is the projective space \({\mathbb{P}}^{m-1}\). We write \(I(\ell, m) = \{\Gamma_1, \dots, \Gamma_{m}\}\), where \(\Gamma_i = (1, 2, \dots, i-1, \widecheck{i}, i+1, \dots, m)\) for \(i = 1, \dots, m\). That is, each
\(\Gamma_i\) is obtained by writing down the string of consecutive integers \(1, \dots, m\) and dropping the integer \(i\). Also, due to our standing
convention, we do not need to consider the Schubert subvariety corresponding to \(\Gamma_m = (1, \dots, m-1)\). Let us now proceed to prove part (a) above and fix \({\alpha}_i\). Note that,
with respect to a fixed basis \(\mathcal{B}\) of \(V_m\), \(\Omega_{\Gamma_i} (m-1, V_m, \mathcal{B})\) is a linear subspace of \({\mathbb{P}}^{m-1}\) of codimension \((i-1)\). Consequently, \(|\Omega_{\Gamma_i} (m-1, V_m, \mathcal{B})| = 1 + \dots + q^{m-i}\). Clearly, a hyperplane of
\({\mathbb{P}}^{m-1}\), that does not contain \(\Omega_{\Gamma_i} (m-1, V_m, \mathcal{B})\) intersect it at most at \(1 + \dots + q^{m-i - 1}\) many points.
This proves part (a). Moreover, part (b) is trivially true since any hyperplane of \({\mathbb{P}}\left(\bigwedge^{m-1} V_m\right)\) is decomposable.
Theorem 15. Let \(q > q_0(\ell)= \frac{2^{\frac{1}{\ell - 1}}}{2^{\frac{1}{\ell - 1}} - 1}\) and \(\ell,m\) be positive integers satisfying \(2 \le \ell < m\). For \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell) \in I(\ell, m)\) and any hyperplane in \({\mathbb{P}}^{k_{\alpha}- 1}\), we have \[|\Pi_{\alpha}\cap \Omega_{\alpha}(\ell, V_m, \mathcal{B})| \le e_{\alpha}(\ell, m) := |\Omega_{\alpha}(\ell, V_m, \mathcal{B})| - q^{\delta({\alpha})}.\] Moreover, the bound is attained if and only if \(\Pi_{\alpha}\) is Schubert Decomposable.
Proof. We prove the assertions by induction on \(m\). Note that the hypothesis mandates \(m \ge 3\). When \(m=3\), we have \((\ell, m) = (2, 3)\), and the assertions follow from the remark above. We may now assume that \(m > 3\) and that both assertions are true for Schubert subvarieties \(\Omega _{\alpha}(\ell, V_{m-1}, \mathcal{B})\) for any vector space \(V_{m-1}\) of dimension \(m-1\) and \(\ell \le m-1\). Let
\(V_m\) be a vector space of dimension \(m\) over \({\mathbb{F}}_q\) with a fixed ordered basis \(\mathcal{B}= \{v_1, \dots,
v_m\}\). For \({\alpha}= ({\alpha}_1, \dots, {\alpha}_\ell) \in I(\ell, m)\), we consider the Schubert subvariety \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\). In view of Remark 14, we may assume that \({\alpha}_1, \dots, {\alpha}_\ell\) are not completely consecutive. If \(\ell =
m-1\), then the assertion follows from Remark 14 (II). In fact, the remark shows that the result is true even without the hypothesis that \(q > q_0 (\ell)\). Thus, we may assume that \(2 \le \ell \le m-2\).
Suppose \(\Pi_{\alpha}\) is a hyperplane of \({\mathbb{P}}^{k_{\alpha}- 1}\). We distinguish the proof in two cases:
Suppose \(\Pi_{\alpha}\) does not contain \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\) for any \(\Omega_{{\alpha}'} (\ell, W_{m-1},
\mathcal{B}_1) \in \mathfrak{S}_{\alpha}(\ell, V_m, \mathcal{B})\). In view of Remark 12 (b) and induction hypothesis, we have, for \(q > q_0
(\ell)\)\[|\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \cap \Pi_{\alpha}| \le |\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)| - q^{\delta ({{\alpha}'})}.\] Since this is true for all elements
of \(\mathfrak{S}_{\alpha}(\ell,V_m,\mathcal{B})\), we may assume that \(M_{{\alpha}} \le |\Omega_{{\alpha}'} (\ell, W_{m-1} , \mathcal{B}_1)| - q^{\delta ({{\alpha}'})}\) as in
Lemma 4 and obtain \[|\Omega_\alpha (\ell, V_{m}, \mathcal{B})\cap \Pi_{\alpha}| \le \left(\frac{q^{m-{\alpha}_k} - 1}{q^{(m - {\alpha}_k) - (\ell - k)} -
1}\right)\left(|\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)| - q^{\delta ({{\alpha}'})}\right).\] Since \(q > q_{0} (\ell)\), using Lemma 5, we have \[|\Omega_\alpha (\ell, V_{m}, \mathcal{B})\cap \Pi_{\alpha}| < |\Omega_\alpha (\ell, V_{m}, \mathcal{B})| - q^{\delta ({\alpha})},\] as desired. We note that, the equality does not
occur in this case.
Suppose \(\Pi_{\alpha}\) contains \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\) for some element in \(\mathfrak{S}_{\alpha}(\ell,V_m,\mathcal{B})\). Proposition 11 applies, and yields \[|\Omega_\alpha (\ell, V_{m}, \mathcal{B})\cap
\Pi_{\alpha}| \le |\Omega_{{\alpha}'}(\ell, m-1, \mathcal{B}_1)| + q^{m-\ell} e_{\widecheck{{\alpha}}}(\ell - 1, m-1).\] Since, by assumption \(q > q_0 (\ell)\) and \(q_0 (\ell) >
q_0 (\ell - 1)\), the induction hypothesis implies that \(e_{\widecheck{{\alpha}}}(\ell - 1, m-1) = |\Omega_{\widecheck{a}} (\ell - 1, W_{m-1}, \mathcal{B}_1')| - q^{\delta (\widecheck{{\alpha}})}\). The desired
inequality now follows from Proposition 9 (a).
Note that, a possible equality arises if and only if \(\Pi_{\alpha}\) contains an element \(\Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) \in \mathfrak{S}_{\alpha}(\ell, V_m,
\mathcal{B})\) and that for each string \(s^{-1} (\nu)\) of \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\setminus \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1)\) the equality
\(|s^{-1}(\nu) \cap \Pi_{\alpha}| = e_{\widecheck{{\alpha}}} (\ell - 1, m-1)\) holds. As noted in Remark 12, we have \(\Omega_\alpha (\ell, V_{m}, \mathcal{B})\setminus \Omega_{{\alpha}'} (\ell, W_{m-1}, \mathcal{B}_1) = \Omega_{{\alpha}} (\ell, V_m, \tilde{\mathcal{B}}) \setminus \Omega_{{\alpha}'} (\ell, W_{m-1},
\tilde{\mathcal{B}}')\). Thus, we may use the bijection \(s^{-1} (\nu) \longleftrightarrow \Omega_{\widecheck{{\alpha}}} (\ell, W_{m-1}, \tilde{B}')\) as discussed before. By induction hypothesis, \(\Pi_{\widecheck{{\alpha}}}\) is Schubert decomposable and hence decomposable.
We claim that, the hyperplanes \(\Pi_{\alpha}\) and \(\Pi_{\widecheck{{\alpha}}}\) are given by the same element of \({\mathbb{P}}\left(\bigwedge^{m-\ell}
V_m\right)\). Note that if the hyperplane \(\Pi_{\alpha}\) is given by the polynomial \(F_{\alpha}= \sum_{\beta \in \nabla ({\alpha}) \cap \Delta ({\alpha}')} c_\beta X_{\beta}\),
then in terms of \((m-\ell)\)-wedges, the polynomial \(F_{\alpha}\) can be rewritten as \[F_{\alpha}= \displaystyle{\sum_{\beta \in \nabla ({\alpha}) \cap \Delta
({\alpha}')}} c_{\beta} v_{\beta^{\mathsf{c}}},\] where for every \(\beta =(\beta_1, \dots, \beta_\ell) \in \nabla ({\alpha}) \cap \Delta ({\alpha}')\), we have \(v_{\beta^{\mathsf{c}}} = v_{\beta_1}' \wedge \cdots \wedge v_{\beta_{m - \ell}'}\) such that \(\{\beta_1, \dots, \beta_\ell\} \cup \{\beta_{1}', \dots, \beta_{m-\ell}'\} = \{1, \dots,
m\}\). Since, for all the \(\beta \in \nabla ({\alpha}) \cap \Delta ({\alpha}')\), we have \(\beta_\ell = m\), we see that \(\beta_j' <
m\), for all \(j = 1, \dots, m-\ell\). On the other hand, given a hyperplane \(\Pi_{\alpha}\) defined by the polynomial \(F_{\alpha}\), as above, the
hyperplane \(\Pi_{\widecheck{{\alpha}}}\) is given by \(F_{\widecheck{{\alpha}}} = \displaystyle{\sum_{\beta \in \nabla ({\alpha}) \cap \Delta ({\alpha}')}} c_\beta
X_{\widecheck{\beta}}\), where for each \(\beta = (\beta_1, \dots, \beta_\ell) \in \nabla ({\alpha}) \cap \Delta ({\alpha}')\), we have \(\widecheck{\beta} = (\beta_1, \dots, \beta_{\ell
- 1})\). At any rate, in terms of \((m-1)-(\ell -1)\) wedges, we may write \[F_{\widecheck{{\alpha}}} = \displaystyle{\sum_{\beta \in \nabla ({\alpha}) \cap \Delta ({\alpha}')}}
c_{\beta} v_{\widecheck{\beta^{\mathsf{c}}}},\] where, as above, for \(\widecheck{\beta} = (\beta_1, \dots, \beta_{\ell - 1}) \in I(\ell - 1, m-1)\), we have \(v_{\widecheck{\beta^{\mathsf{c}}}} = v_{\beta_1'} \wedge \dots \wedge v_{\beta_{m-\ell}'}\) such that \(\{\beta_1, \dots, \beta_{\ell - 1}\} \cup \{\beta_1', \dots, \beta_{m-\ell}'\} =
\{1, \dots, m-1\}\). Thus our claim is proved.
Thus the decomposability of \(\Pi_{\widecheck{{\alpha}}}\) guarantees that \(\Pi_{\alpha}\) is also decomposable. Thus by [7], as pointed out in Remark 5 (a), that \(\Pi_{\alpha}\) is Schubert decomposable. ◻
We conclude this article with a remark on the quantity \(q_0 (\ell)\).
Remark 16. The function \(q_0 (\ell)\), as pointed out before, is an increasing function on \(\ell\). One can check with a calculator that \(q_0 (2) = 2\), \(q_0 (3) \approx 3.14, ...\). In particular, our result establishes the MWCC for \(\ell = 2\) with \(q \ge 3\),
\(\ell = 3\) with \(q \ge 4\) etc. We also note, and leave it to the reader to check that, \[\lim_{\ell \to \infty} \dfrac{q_0 (\ell)}{\ell} = \frac{1}{\ln 2}
\approx 1.4427.\] An obvious question arises. How can one improve the situation? In particular, is it possible to use our methodology to prove the MWCC in complete generality? If not, is it at least possible to replace \(q_0 (\ell)\) with some \(q_1 (\ell)\) in our hypothesis of Theorem 15 where \(q_1 (\ell) <
q_0(\ell)\) for all large values of \(\ell\)? We do not yet have answers to these questions. However, it is conceivable that the possible improvement lies in obtaining better bounds for the number of points on
Schubert subvarieties over finite fields in Proposition 6, which might also lead to understanding the quantities \({\alpha}_i\) in ?? . We leave these as
problems for further research.
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The first named author is partially supported by a research grant 02011/34/2025/R&D-II/DAE/9465 from the National Board for Higher Mathematics, Department of Atomic Energy, India.↩︎
The second named author is partially supported by a DST-INSPIRE Ph.D. fellowship from the Department of Science and Technology, Govt. of India↩︎
The third named author was supported by the Tromsø Research Foundation (project “Pure Mathematics in Norway”) and the UiT Aurora project MASCOT.↩︎