1 Introduction↩︎

The goal of this paper is to introduce an explicit algebraic counterpart of the geometric Pappas–Rapoport coherence conjecture in type A (Zhu’s theorem) [1], [2]. The input of our construction is a collection of cyclic representations of the current algebra, the elements of the collection are labeled by the simple roots of the underlying affine Kac–Moody Lie algebra. The output is a (non-cyclic) representation of the Iwahori algebra; the character of this representation coincides with that of the Cartan component of the tensor product of the initial representations. If one starts with a very special collection of spherical Demazure submodules of integrable irreducible representations, whose highest weights are multiples of a fixed level-one weight, then the resulting Iwahori representation is a sum of Demazure submodules in another integrable irreducible representation. The input and output in this case are exactly the left- and right-hand sides of the coherence conjecture of Pappas and Rapoport. Let us provide more details.

Throughout the paper, we work with the Lie algebras \(\mathfrak{gl}_n\) and \(\mathfrak{sl}_n\), and their affine Kac–Moody algebras [3]. Let \({\mathcal{G}r}\) be the affine Grassmannian for the adjoint group of \(\mathfrak{sl}_n\); in particular, \({\mathcal{G}r}\) is the disjoint union of \(n\) components \({\mathcal{G}r}_b\), where \(b=0,\dots,n-1\) [3], [4]. The ind-varieties \({\mathcal{G}r}_b\) are covered by spherical Schubert varieties of the form \(X_\lambda\), where \(\lambda\) are partitions with at most \(n\) parts. Each \(X_\lambda\) is acted upon by the current group \(SL_n[[z]]\), which is a subgroup of the affine Kac–Moody Lie group \(\widehat{SL}_n\) acting on \({\mathcal{G}r}_b\) (in fact, \({\mathcal{G}r}_b\) are partial flag varieties for \(\widehat{SL}_n\)).

Let \(\EuScript{O}(1)\) be the ample line bundle on \({\mathcal{G}r}_b\), generating the corresponding Picard group. Then the restricted dual of the space of sections \(H^0({\mathcal{G}r}_b,\EuScript{O}(1))\) is identified with the integrable irreducible highest weight representation \(L(\Lambda_b)\) of the affine Kac–Moody Lie algebra \(\widehat\mathfrak{sl}_n\) and one has an embedding \({\mathcal{G}r}_b\subset {\mathbb{P}}(L(\Lambda_b))\). We note that for any \(k\in{\mathbb{Z}}_{>0}\), the highest weight \(\widehat\mathfrak{sl}_n\) module \(L(k\Lambda_b)\) is identified with the dual space of sections of the line bundle \(\EuScript{O}(k)=\EuScript{O}(1)^{\otimes k}\). Now let us restrict \(\EuScript{O}(k)\) to \(X_\lambda\). Then \(H^0(X_\lambda,\EuScript{O}(k))^*\) admits an action of the current algebra \(\mathfrak{sl}_n[z]\) and is isomorphic to the Demazure module \(D_{k,\lambda}\subset L(k\Lambda_b)\), which is a cyclic representation of \(\mathfrak{sl}_n[z]\) with a cyclic vector of weight \(k\lambda\).

The affine Grassmannians \({\mathcal{G}r}_b\) admit degeneration to the complete flag variety \({\mathcal{F}l}\). More precisely, there exists a family over \({\mathbb{A}}^1\) whose general fiber is isomorphic to \({\mathcal{G}r}_b\) and the special fiber is isomorphic to \({\mathcal{F}l}\) (see [5]–[8]). The family is defined for arbitrary groups, but in type \(A\) it admits a very concrete description in terms of lattices [7], [9]. If one considers a subfamily whose general fiber is the Schubert variety \(X_\lambda\), then the special fiber is identified with a union of Schubert subvarieties \(Y_w\) inside the affine flag variety \({\mathcal{F}l}\) [2], [10], [11]. The varieties \(Y_w\) are labeled by the elements of the so-called admissible set \(\mathcal{A}_\lambda\), which is a cardinality \(|W|\) subset of the extended affine Weyl group (\(W=S_n\) is the finite Weyl group). We note that \(Y_w\) are, in general, not invariant with respect to the whole current algebra, but admit an action of the Iwahori subalgebra \({\mathfrak I}\).

The Picard group of the affine flag variety has a collection of generators \(\EuScript{L}_b\), \(b=0,\dots,n-1\), where \(\EuScript{L}_b\) is the pullback of \(\EuScript{O}(1)\) with respect to the natural projection map \({\mathcal{F}l}\to{\mathcal{G}r}_b\). In particular, for any collection of non-negative integers \({\boldsymbol{k}}=(k_b)_b\), the space of sections of the line bundle \(\EuScript{L}_{\boldsymbol{k}}=\bigotimes_{b} \EuScript{L}_b^{\otimes k_b}\) is identified with the dual of the integrable highest weight \(\widehat{\mathfrak{sl}}_n\) module \(L(\Lambda_{\boldsymbol{k}})\), where the weight \(\Lambda_{\boldsymbol{k}}\) is equal to \(\sum k_b\Lambda_b\), \(\Lambda_b\) are the affine fundamental weights. The level of the weight \(\Lambda_{\boldsymbol{k}}\) is equal to \(|{\boldsymbol{k}}|=\sum k_b\). The Demazure modules \(D_w(\Lambda_{\boldsymbol{k}})\subset L(\Lambda_{\boldsymbol{k}})\) are representations of the Iwahori Lie algebra \({\mathfrak I}\); they are identified with the dual space of sections \(H^0(Y_w,\EuScript{L}_{\boldsymbol{k}})^*\).

The coherence conjecture proved by Zhu states that \[\label{eq:cohconj} \dim H^0\bigl(\bigcup_{w\in\mathcal{A}_\lambda} Y_w,\EuScript{L}_{\boldsymbol{k}}\bigr) = \dim H^0(X_\lambda,\EuScript{O}(|{\boldsymbol{k}}|)).\tag{1}\] The equality of dimensions was recently upgraded in [12] to the equality (up to a certain shift) of characters with respect to the Cartan subalgebra (in fact, as shown in [12], a larger algebra may show up). Let us emphasize that the left-hand side of 1 is a representation of the Iwahori algebra and the right-hand side is a representation of the current algebra. Both the Zhu proof [2] and the Hong–Yu construction use line bundles on the global affine Grassmannian: the right-hand side of 1 is the space of sections on the general fiber, and the left-hand side is the space of sections on the special fiber. Hence, 1 can be seen as a degeneration of the space in the right-hand side to the space in the left-hand side. Our main goal is to give an algebraic description of this degeneration procedure. To this end, we introduce a more general construction which leads to the desired description in certain special cases.

Our construction starts with a collection of \(n\) cyclic representations \(D_b\), \(b=0,\dots,n-1\) of \(\mathfrak{sl}_n[z]\) with cyclic vectors \(d_b\). The output is a (no longer cyclic) representation \({\boldsymbol{D}}(0)\) of the Iwahori algebra. The character of \({\boldsymbol{D}}(0)\) is equal to the character of the Cartan component \(D\) inside the tensor product of the modules \(D_b\): \(D = \mathrm U(\mathfrak{sl}_n[z]).\bigotimes d_b\). The module \({\boldsymbol{D}}(0)\) can be seen as a degeneration of \(D\). Let us describe our construction in detail for \(D_b\) being irreducible \(\mathfrak{sl}_n\) modules with the trivial action of \(z\mathfrak{sl}_n[z]\) (for the general case see section 7).

We start with an integral dominant \(\mathfrak{sl}_n\) weight \(\lambda\) and a decomposition \(\lambda=\sum_b \lambda^{(b)}\) into a sum of \(n\) weights \(\lambda^{(b)}\). Then the irreducible \(\mathfrak{sl}_n\) module \(V_\lambda\) sits inside the tensor product of \(V_{\lambda^{(b)}}\) as a Cartan component. Let \(v_b\in V_{\lambda^{(b)}}\) be the lowest weight cyclic vectors, so each module is generated from \(v_b\) by the action of the Chevalley generators \(e_i\in\mathfrak{sl}_n\), \(i=1,\dots,n-1\) corresponding to the simple roots. We deform \(V_\lambda\) inside the tensor product \(\bigotimes_b V_{\lambda^{(b)}}\). The deformation is defined via the operators \(e_i(\varepsilon)= \varepsilon e_i^{(i)}+\sum_{b\ne i} e_i^{(b)}\), where \(e_i^{(b)}\) is the operator acting as \(e_i\) at the \(b\)-th factor of the tensor product and as identity on all other factors. Then we define the space \(V_{\bar\lambda}(\varepsilon)\) inside \(\bigotimes_b V_{\lambda^{(b)}}\) as the subspace generated from the tensor product of lowest weight vectors \(v_b\) by the action of operators \(e_i(\varepsilon)\). Let \(V_{\bar\lambda}=V_{\bar\lambda}(0)\) be the \(\varepsilon\to 0\) limit of the spaces \(V_{\bar\lambda}(\varepsilon)\). We show that \(V_{\bar\lambda}\) carries a natural action of (certain quotient of) the Iwahori algebra \({\mathfrak I}\).

We are not able to prove the conjecture, but we have performed numerous checks using the program found in Appendix 9. We note that the Iwahori algebra acts on \(V_{\bar\lambda}\) via a certain quotient, called Inonu–Wigner contraction (see [13], [14]), which is a close cousin of the Drinfeld double of the Borel subalgebra [15]–[17]. This quotient can also be realized in two other ways: as a degeneration of the Lie algebra \(\mathfrak{sl}_n\) [18]–[20], and as an endomorphism algebra of certain representation of the cyclic equioriented quiver [21]–[23].

In order to describe the \({\mathfrak I}\) modules \(V_{\bar\lambda}\) in general, we consider the Kostant–Kumar modules (see [24]–[26]). Let us fix a collection \(\bar\Lambda\) of affine integrable weights \((\Lambda_b)_b\) and a collection \(\bar w\) of affine Weyl group elements \((w_b)_b\); let \(d_b\in L(\Lambda_b)\) be the extremal vectors of weight \(w_b(\Lambda_b)\). We define \(K({\bar w},\bar\Lambda)\subset \bigotimes_b L(\Lambda_b)\) as \(\mathrm U({\mathfrak I}).\bigotimes_b d_b\).

Let \(V_{\bar\lambda}'\subset V_{\bar\lambda}\) be the Iwahori algebra subrepresentation inside the tensor product of \(V_{\lambda^{(b)}}\) generated from the tensor product of extremal vectors (if the conjecture above is true, then \(V_{\bar\lambda}'=V_{\bar\lambda}\)). We show that the \({\mathfrak I}\)-module \(V_{\bar\lambda}'\) is isomorphic to a direct sum of \(|W|\) many affine Kostant–Kumar modules [24], [27].

Let us close with two remarks. First, the algebraic degeneration of \(V_{\sum \lambda^{(b)}}\) to \(V_{\bar\lambda}\) has a geometric counterpart: the flag variety degenerates to a union of affine Kostant–Kumar Schubert varieties. We expect that Conjecture 2 is related to the flatness of this degeneration. Second, all the constructions above work well in the generality of arbitrary spherical Demazure modules \(D_{\lambda}\) instead of the irreducible \(\mathfrak{sl}_n\) modules \(V_\lambda\); the corresponding formalism can be found in section 7. However, the corresponding Iwahori algebra module is, in general,not generated from the extremal weight vectors (see examples and non-examples in section 8).

The paper is organized as follows. In Section 2, we collect main finite-dimensional and infinite-dimensional objects used in the paper. In Section 3, we formulate the Zhu theorem proving the Pappas–Rapoport coherence conjecture. In Section 4, the lattice formalism for the type A affine Grassmannians and flag varieties is recalled. Section 5 contains the main construction of our paper, which produces an Iwahori algebra module starting from a collection of spherical Demazure modules. In Section 6, we consider the special case where the Demazure modules are irreducible finite-dimensional representations of \(\mathfrak{sl}_n\) and in Section 7 we treat a more general case. Section 8 contains several explicit examples of our general construction. Finally, in Appendix 9, we present the computer program used to verify special cases of Conjecture 2.

Acknowledgments↩︎

EF and AK were partially supported by ISF grant 493/24.

2 Preliminaries↩︎

2.1 Finite-dimensional algebras↩︎

For \({\mathfrak g}=\mathfrak{sl}_n\) let us fix the Cartan decomposition \({\mathfrak g}={\mathfrak b}\oplus{\mathfrak n}_-\), \({\mathfrak b}=\mathfrak{h}\oplus{\mathfrak n}\), where \(\mathfrak{h}\) is the diagonal Cartan subalgebra and \({\mathfrak n}\) (resp., \({\mathfrak n}_-\)) are the upper- (resp., lower-) triangular subalgebras. Let \(\Phi=\Phi_+\sqcup\Phi_-\subset \mathfrak{h}^*\) be the set of roots decomposed into the disjoint union of positive and negative roots. We denote by \(\alpha _i\in\Phi_+\), \(i=1,\dots,n-1\) the simple roots for \({\mathfrak g}\) and by \(f_i=f_{\alpha _i}\in{\mathfrak n}_-\), \(e_i=e_{\alpha _i}\in{\mathfrak n}\) the corresponding Chevalley generators. Explicitly, \(f_i=E_{i+1,i}\), \(e_i=E_{i,i+1}\) where \(E_{\bullet,\bullet}\) are matrix units. For a positive root \(\alpha\) we denote by \(f_\alpha \in{\mathfrak n}_-\) the Cartan generator of weight \(-\alpha\) and by \(e_\alpha \in{\mathfrak n}\) the Chevalley generator of weight \(\alpha\). The highest root \(\theta\) is equal to the sum of all simple roots \(\alpha _i\). The Weyl group \(W\simeq S_n\) is generated by simple reflections \(s_i\), \(i=1,\dots,n-1\) corresponding to simple roots \(\alpha _i\); we denote by \(w_0\) the longest element in \(W\).

Let \(\omega _i\), \(i=1,\dots,n-1\) be the fundamental weights and let \(P\) be the weight lattice containing the cone of integral fundamental weights \(P_+=\bigoplus_i {\mathbb{Z}}_{\ge 0} \omega _i\). The lattice \(P\) contains the root lattice \(Q\) generated by the simple roots \(\alpha _i\). Let \((\cdot,\cdot)\) be the standard invariant Killing form on \(\mathfrak{h}^*\). The simple coroots \(\alpha _i^\vee\in\mathfrak{h}\) are defined by \((\alpha _i,\beta)=\beta(\alpha_i^\vee)\). The simple coroots \(\alpha _i^\vee\) generate the coroot lattice \(Q^\vee\) and the fundamental coweights \(\omega _i^\vee\) generate the coweight lattice \(P^\vee\).

The irreducible highest weight modules of \(\mathfrak{sl}_n\) are denoted by \(V_\mu\) with \(\mu\in P_+\). In each \(V_\mu\) we fix a highest weight vector and denote it by \(v_\mu\). The extremal weights of \(V_\mu\) are the \(W\)-shifts \(\sigma\mu\) of \(\mu\); for each \(\sigma\in W\) let \(v_{\sigma\mu}\in V_\mu\) be a fixed extremal weight vector spanning its weight space. The Demazure modules \(D_{\sigma\mu}\subset V_\mu\) are defined as \(\mathrm U({\mathfrak b})v_{\sigma\mu}\). In particular, \(D_{w_0\mu}=V_\mu\) and \(D_\mu\) is a one-dimensional space spanned by the highest weight vector \(v_\mu\).

In what follows we work with both the special linear Lie algebra \(\mathfrak{sl}_n\) and the general linear Lie algebra \(\mathfrak{gl}_n\). The Cartan (diagonal) algebra of \(\mathfrak{gl}_n\) if \(n\)-dimensional; the coweights (in the standard basis) are denoted by \(\lambda=(\lambda_1,\dots,\lambda_n)\), in particular, the compositions \(\lambda\) form the lattice of integral coweights. Let \({\mathbb{X}}\simeq {\mathbb{Z}}^n\) be coweight lattice; we denote by \(\omega _i^\vee\in{\mathbb{X}}\) the elements given by \((1,\dots,1,0,\dots,0)\) (with \(i\) units). Hence every fundamental coweight of \(\mathfrak{sl}_n\) can be seen inside \({\mathbb{X}}\) and we get an embedding of the \(\mathfrak{sl}_n\) coweight lattice \(P^\vee\) into \({\mathbb{X}}\). In particular, the image of the dominant cone \(P^\vee_+\) is identified with the set of integral dominant coweights \(\lambda\) (\(\lambda_i\ge \lambda_{i-1}\) for all \(i\)) such that \(\lambda_n=0\).

Finally, let us note that the irreducible highest weight \(\mathfrak{gl}_n\) modules \(V_\mu\) are labeled by dominant compositions \(\mu\); we note that the overall shift of all components of \(\mu\) does not change \(V_\mu\) as a module over the embedded \(\mathfrak{sl}_n\subset\mathfrak{gl}_n\).

2.2 Infinite-dimensional algebras↩︎

For a Lie algebra \({\mathfrak g}\) the corresponding current algebra \({\mathfrak g}[z]\) is defined as \({\mathfrak g}[z]={\mathfrak g}\otimes{\mathbb{C}}[z]\). It contains a finite-dimensional subalgebra \({\mathfrak g}\simeq {\mathfrak g}\otimes z^0\) and an infinite-dimensional Iwahori subalgebra \({\mathfrak I}={\mathfrak b}[z]\oplus z{\mathfrak n}_-[z]\). The Iwahori algebra is generated by the elements \(f_i=f_i\otimes 1\), \(i=1,\dots,n-1\), by the element \(f_0=e_\theta\otimes z\) and by the Cartan subalgebra.

The Iwahori subalgebra plays a role of a Borel subalgebra inside the affine Kac–Moody Lie algebra \(\widehat\mathfrak{g}={\mathfrak g}\otimes{\mathbb{C}}[z,z^{-1}]\oplus{\mathbb{C}}K\oplus{\mathbb{C}}d\), where \(K\) is central and \(d\) is the degree operator. The affine Cartan subalgebra \(\mathfrak{h}^a\subset \widehat\mathfrak{g}\) is spanned by \(\mathfrak{h}\otimes 1\) and the elements \(K\), \(d\). The dual Cartan subalgebra \((\mathfrak{h}^a)^*\) is spanned by the roots \(\alpha _i\), \(i=1,\dots,n-1\) (which vanish at \(K\) and \(d\)) and the elements \(\Lambda_0\), \(\delta\) such that \(\Lambda_0(\alpha _i)=\delta(\alpha _i)=0\) for \(1\le i<n\), \(\Lambda_0(K)=1=\delta(d)\), \(\Lambda_0(d)=\delta(K)=0\). The affine simple root \(\alpha _0\) is equal to \(\delta-\theta\).

An affine dominant integral weight \(\Lambda\in(\mathfrak{h}^a)^*\) is defined by its restriction to the finite part \(\Lambda|_\mathfrak{h}\) and by the level \(\Lambda(K)\) (we will always assume that \(\Lambda(d)=0\)). The level is a non-negative integer and one has the restriction \((\Lambda|_\mathfrak{h},\theta^\vee)\le \Lambda(K)\). For \({\mathfrak g}=\mathfrak{sl}_n\) there are \(n\) level one integral dominant weights: the possible finite parts are fundamental weights or zero. We denote the lattice of affine dominant integral weights of \(\widehat{\mathfrak{sl}}_n\) by \(P^a_+\) and the representation corresponding to \(\Lambda\in P^a_+\) by \(L(\Lambda)\). The space \(L(\Lambda)\) contains a highest weight vector \(v_\Lambda\) such that \(L(\Lambda)\) is generated from \(v_\Lambda\) by the action of \({\mathfrak b}_-[z^{-1}]\oplus z^{-1}{\mathfrak n}[z^{-1}]\).

The affine type \(A^{(1)}_{n-1}\) Dynkin diagram (corresponding to the Lie algebra \(\widehat\mathfrak{sl}_{n}\)) is the cyclic graph \(\Delta\) with \(n\) vertices. We denote the set of vertices by \(\Delta_0\) and identify \(\Delta_0\) with the set of numbers \(b\in{\mathbb{Z}}/n{\mathbb{Z}}\); we also assume that the graph \(\Delta\) is equi-oriented. Let \(W^a\) be the affine Weyl group. The group \(W^a\) is generated by the simple reflections \(s_b\), \(b\in\Delta_0\). It can be explicitly written as \(W^a=W\ltimes Q^\vee\), where \(Q^\vee\) is the coroot lattice. For \(\lambda\in Q^\vee\) we denote the corresponding element of \(W^a\) by \(z^\lambda\).

For \(b\in \Delta_0\) let \(\Lambda_b\in(\mathfrak{h}^a)^*\) be the affine level one fundamental weights: the finite part of \(\Lambda_0\) is zero, \(\Lambda_b|_\mathfrak{h}=\omega_b\) for \(b\ne 0\). The weight \(\Lambda_0\) is called basic (or vacuum) level one weight. Any integral dominant weight \(\Lambda\) can be written as \(\sum_{b\in\Delta_0} m_b\Lambda_b\) with some non-negative integer coefficients \(m_b\).

For an element \(\sigma\in W^a\) let \(v_{\sigma\Lambda}\in L(\Lambda)\) be a fixed extremal vector of weight \(\sigma\Lambda\). The affine Demazure module \(D_\sigma(\Lambda)\subset L(\Lambda)\) is equal to \(\mathrm U({\mathfrak I})v_{\sigma\Lambda}\), i.e. is generated from the extremal weight vector by the action of the Iwahori algebra.

We also need the \(\mathfrak{sl}_n\) and \(\mathfrak{gl}_n\) extended affine Weyl groups. Recall that the type \(A_{n-1}^{(1)}\) extended affine Weyl group is the semi-direct product of the finite Weyl group \(W\) and the coweight lattice \(P^\vee\). The \(\mathfrak{gl}_n\) version is the semi-direct product \(W_{\rm ext}=S_n\ltimes {\mathbb{X}}\).

3 The coherence conjecture↩︎

Recall the notation \({\mathbb{K}}={\mathbb{C}}((z))\) and \({\mathbb{O}}={\mathbb{C}}[[z]]\) for the field of formal Laurent series and its subring of formal power series. For a simple Lie algebra \({\mathfrak g}\) let \(G\) be the corresponding simply-connected Lie group; in this paper we only consider the case \({\mathfrak g}=\mathfrak{sl}_n\), \(G=SL_n\). For \(b\in\Delta_0={\mathbb{Z}}/n{\mathbb{Z}}\) let \({\mathcal{G}r}_b\) be the corresponding affine Grassmannian, which is the quotient of \(G({\mathbb{K}})\) by the \(b\)-th maximal parabolic subgroup. In particular, \({\mathcal{G}r}_0\) is the quotient of \(G({\mathbb{K}})\) by the current group \(G({\mathbb{O}})=G[[z]]\). All the affine Grassmannians are ind-varieties, i.e. inductive limits of finite-dimensional subvarieties. One has a \(\widehat G\)-equivariant embedding \({\mathcal{G}r}_b\subset{\mathbb{P}}(\Lambda_b)\); we denote by \(\EuScript{O}\) the pull-back from \({\mathbb{P}}(\Lambda_b)\) of the line bundle \(\EuScript{O}(1)\). The line bundle \(\EuScript{O}\) is known to be a generator of the Picard group of the affine Grassmannian \({\mathcal{G}r}_b\).

The Schubert varieties inside affine Grassmannians \({\mathcal{G}r}_b\) are defined as the closures of the Iwahori group orbits through the torus fixed points. We will only be interested in the spherical Schubert varieties in the affine Grassmannians, i.e. whose Schubert varieties which are invariant with respect to the action of the current group \(SL_n({\mathbb{O}})\). The spherical Schubert varieties are labeled by dominant coweights \(\lambda\); the corresponding Schubert variety is denoted by \(X_\lambda\) (since we are working in type \(A\), we sometimes identify coweight lattice with the weight lattice). For a fixed \(\lambda\) the variety \(X_\lambda\) sits inside \({\mathcal{G}r}_b\) such that \(\lambda-\omega _b^\vee\) is in the coroot lattice (if \(\lambda\) itself belongs to the coroot lattice, then \(b=0\)).

Let \({\mathcal{F}l}=G({\mathbb{K}})/{\boldsymbol{I}}\) be the affine flag variety, where \({\boldsymbol{I}}\) is the Iwahori subgroup – the preimage of the Borel subgroup of \(G\) with respect to the \(z=0\) evaluation map. As in the case of affine Grassmannians, \({\mathcal{F}l}\) is an ind-variety as an inductive limit of finite-dimensional Schubert subvarieties (for more details see below). Using the projections \({\mathcal{F}l}\to {\mathcal{G}r}_b\), one constructs line bundles \(\EuScript{L}(\Lambda_b)\) on \({\mathcal{F}l}\) as pull backs of \(\EuScript{O}\). The line bundles \(\EuScript{L}(\Lambda_b)\), \(b\in\Delta_0\) generate the Picard group of the affine flag variety. For \(\Lambda=\sum_b m_b\Lambda_b\) we denote by \(\EuScript{L}(\Lambda)\) the line bundle \(\bigotimes_{b\in\Delta_0} \EuScript{L}(\Lambda_b)^{\otimes m_b}\). One can also define partial affine flag varieties, interpolating between \({\mathcal{F}l}\) and \({\mathcal{G}r}_b\), but in this paper we need only the complete affine flags.

The affine flag variety contains the set of torus fixed points \(p_\sigma\) for \(\sigma\in W^a\). The Iwahori group orbits \({\boldsymbol{I}}.p_\sigma\) are called Schubert cells and the closures are called Schubert varieties. We denote \(\overline{{\boldsymbol{I}}.p_\sigma}\) by \(Y_\sigma\subset {\mathcal{F}l}\). Each \(Y_\sigma\) is a finite-dimensional (in general, singular) projective algebraic variety; the union of all Schubert varieties coincides with \({\mathcal{F}l}\). One has the following important property of the Schubert varieties: the space of sections \(H^0(Y_\sigma,\EuScript{L}(\Lambda))\) is isomorphic (as the Iwahori algebra module) to the dual of an affine Demazure module.

In order to formulate the Zhu theorem (Pappas–Rapoport conjecture) we need one more piece of notation. For a coweight \(\lambda\) let \(z^\lambda\) be the corresponding extended affine Weyl group element. Let \[\label{eq:Ala} \mathcal{A}(\lambda) = \bigcup_{w\in W} Y_{z^{w\lambda}}\subset {\mathcal{F}l},\tag{2}\] i.e. \(\mathcal{A}(\lambda)\) is a union of (closed) Schubert varieties inside the affine flag variety.

Let \(\Lambda=\sum_{b\in\Delta_0} m_b \Lambda_b\) be an affine dominant integral weight. In particular, the level \(\Lambda(K)\) of \(\Lambda\) is equal to the sum of all \(m_b\). Recall the line bundles \(\EuScript{L}(\Lambda)\) on \({\mathcal{F}l}\) and the line bundle \(\EuScript{O}\) on the affine Grassmannians. The coherence conjecture states that \[\dim H^0(X_\lambda,\EuScript{O}^{\otimes\Lambda(K)}) = \dim H^0(\mathcal{A}(\lambda),\EuScript{L}(\Lambda)).\] The equality of dimensions was upgraded to the isomorphism of modules over the Cartan subalgebra by Hong and Yu (see [12]).

4 Global affine Grassmannians in type A↩︎

4.1 Lattices↩︎

The major role in the proof of the coherence conjecture is played by the global affine Grassmannians, which is a family connecting affine Grassmannians and affine flag varieties (see [2], [12], [28]). In type \(A\) one can make the construction explicit using the lattice formalism (see e.g. [8], [9], [29]). Let us recall the setup.

Let \(w_1,\dots,w_n\) be a standard basis of the \(n\)-dimensional vector space. A lattice \(L\) is a subspace in \({\mathbb{K}}^n=\mathrm{span}\{w_i\}_{i=1}^n\otimes{\mathbb{K}}\) which is a free \({\mathbb{O}}\) module of rank \(n\). For example, for a coweight \(\lambda=(\lambda_1,\dots,\lambda_n)\in{\mathbb{Z}}^n\) we denote by \({\boldsymbol{L}}^\lambda\) the lattice generated by \(z^{-\lambda_i}w_i\), \(1\le i\le n\). The charge \(\nu({\boldsymbol{L}}^\lambda)\) is equal to the sum of all \(\lambda_i\): \(\nu({\boldsymbol{L}}^\lambda)=|\lambda|\). For example, for \(\omega _b=(1,\dots,1,0,\dots,0)\) (with \(b\) units) the lattice \({\boldsymbol{L}}^{\omega _b}\) is equal to \({\mathbb{O}}^n\oplus z^{-1}\mathrm{span}\{w_1,\dots,w_b\}\) and \(\nu({\boldsymbol{L}}^{\omega _b})=b\). For a general lattice \(L\) with an \({\mathbb{O}}\) basis \(u_1,\dots,u_n\) its charge \(\nu(L)\) is defined as the negated smallest \(z\)-degree showing up in the determinant of the matrix whose columns are the \(n\)-vectors \(u_i\) (with coefficients in \({\mathbb{K}}\)). The group \(SL_n({\mathbb{O}})\) acts on the space of lattices and preserves the charge.

The affine Grassmannian \({\mathcal{G}r}_b\) is realized as the space of lattices \(L\) such that \(\nu(L)=b\); in particular, \({\mathcal{G}r}_b\ni {\boldsymbol{L}}^{\omega _b}\) (including the case \(\omega _0=0\)). The affine Grassmannian \({\mathcal{G}r}_b\) is isomorphic to the quotient of \(SL_n({\mathbb{K}})\) by the \(b\)-th maximal parabolic subgroup. In general, all lattices realize the \(GL_n\) affine Grassmannian \(GL_n({\mathbb{K}})/GL_n({\mathbb{O}})\). We note that the multiplication by \(z^{-1}\) adds \(n\) to the charge of a lattice, so in principal one can consider affine Grassmannians \({\mathcal{G}r}_b\) for any integer \(b\), consisting of lattices of charge \(b\), but \({\mathcal{G}r}_b\) is naturally identified with \({\mathcal{G}r}_{b+n}\).

The affine flag variety \({\mathcal{F}l}\) sits inside the product of affine Grassmannians: \({\mathcal{F}l}\subset \prod_{b\in\Delta_0} {\mathcal{G}r}_b\). Explicitly, \({\mathcal{F}l}\) consists of collections \((L_b)_{b}\) such that \[L_0\subset L_1\subset\dots\subset L_{n-1}\subset z^{-1}L_0,\;\dim L_{i+1}/L_i=1.\] In particular, the base point of \({\mathcal{F}l}\) is represented by the chain \(({\boldsymbol{L}}^{\omega _b})_{b=0}^{n-1}\). For \(w=(\sigma,\lambda)\in W_{\rm ext}\), \(\sigma\in S_n\), \(\lambda\in{\mathbb{X}}\) we define the point \({\boldsymbol{L}}^w\in{\mathcal{F}l}\) by \[\label{eq:Lw} {\boldsymbol{L}}^w=\left({\boldsymbol{L}}^{\sigma(\lambda)}\subset {\boldsymbol{L}}^{\sigma(\lambda+\omega _1)}\subset \dots\subset {\boldsymbol{L}}^{\sigma(\lambda+\omega _{n-1})} \subset z^{-1}{\boldsymbol{L}}^{\sigma(\lambda)}\right).\tag{3}\] As mentioned above, the spherical Schubert varieties in the affine Grassmannian are labeled by dominant coweights \(\lambda\); the variety \(X_\lambda\) is the closure of the orbit \(SL_n({\mathbb{O}}).{\boldsymbol{L}}^\lambda\). The spherical Schubert varieties sitting inside \({\mathcal{G}r}_b\) correspond to \(\lambda\) with \(|\lambda|=b\). The Schubert subvarieties inside the affine flag variety \({\mathcal{F}l}\) are the closures of the Iwahori group orbits of the points \({\boldsymbol{L}}^w\), \(w\in W_{\rm ext}\).

4.2 Global affine Grassmannian↩︎

Let us describe a family over \({\mathbb{A}}^1\) whose general fiber is an affine Grassmannian \({\mathcal{G}r}_b\) and the special fiber is isomorphic to the affine flag variety. For simplicity, the family below corresponds to the case \(b=0\), the general case does not differ much.

We introduce the \({\mathbb{K}}\)-linear maps \(\psi_b: {\mathbb{K}}^n\to {\mathbb{K}}^n\), \(b\in{\mathbb{Z}}/n{\mathbb{Z}}\) by \[\label{eq:psib} \psi_b(w_i)=\begin{cases} w_i, & i\ne b+1,\\ (z+\varepsilon)w_i, & i=b+1\end{cases},\tag{4}\] where \(\varepsilon\) is a complex number. The global Grassmannian \(\boldsymbol{Gr}\) is formed by collections \((\varepsilon,L_0,\dots,L_{n-1})\) such that \(\varepsilon\in{\mathbb{C}}\), \(L_i\in{\mathcal{G}r}_0\) and \(\psi_{b} L_i\subset L_{i+1}\) for all \(i=0,\dots,n-1\) (the condition for \(i=n-1\) reads as \(\psi_{n-1} L_{n-1}\subset L_0\)). We note that the composition of all the maps \(\psi_{b}\) is equal to \((z+\varepsilon)\mathrm{Id}\), the condition \((z+\varepsilon)L_0\subset L_0\) does hold, since \(L_0\) is a lattice. The general fiber of the natural map \(\pi:\boldsymbol{Gr}\to{\mathbb{C}}\) (projection to the first coordinate \(\varepsilon\)) is isomorphic to the affine Grassmannian \({\mathcal{G}r}_0\), since \(z+\varepsilon\) is invertible in \({\mathbb{O}}\) for \(\varepsilon\ne 0\). and the special fiber (over \(\varepsilon=0\)) is isomorphic to the affine flag variety \({{\mathcal{F}l}}\); the isomorphism is given by \((L_i)_i\mapsto (A^iL_i)\), where \(A\) is the natural shift operator which identifies \({\mathcal{G}r}_\bullet\) with \({\mathcal{G}r}_{\bullet+1}\).

The spherical Schubert varieties \(X_\lambda\) admit a global version \(\boldsymbol{X}_\lambda\subset \boldsymbol{Gr}\) [8]. The general fiber of the restriction of the projection map \(\pi\) to \(\boldsymbol{X}_\lambda\) is isomorphic to the Schubert variety \(X_\lambda\) and the special fiber \(\pi^{-1}(0)\cap \boldsymbol{X}_\lambda\subset {\mathcal{F}l}\) is equal to a union of Schubert varieties \(\mathcal{A}_\lambda\) 2 . One can describe the components explicitly using the Kottwitz-Rapoport alcoves [30].

5 The construction↩︎

In this section we formulate the general form of the algebraic version of the Pappas–Rapoport construction. In the following sections we discuss the details and describe certain special cases (in particular, related to the coherence conjecture).

Here is an example for \(n=4\): \[A_0= \left(\begin{smallmatrix} a_{11} & a_{12} & a_{13} & a_{14}\\ a_{21} & a_{22} & a_{23} & a_{24}\\ a_{31} & a_{32} & a_{33} & a_{34}\\ a_{41} & a_{42} & a_{43} & a_{44} \end{smallmatrix}\right),\; A_1= \left(\begin{smallmatrix} a_{11} & a_{12}(z+\varepsilon) & a_{13}(z+\varepsilon) & a_{14}(z+\varepsilon)\\ \frac{a_{21}}{z+\varepsilon} & a_{22} & a_{23} & a_{24}\\ \frac{a_{31}}{z+\varepsilon} & a_{32} & a_{33} & a_{34}\\ \frac{a_{41}}{z+\varepsilon} & a_{42} & a_{43} & a_{44} \end{smallmatrix}\right),\] \[A_2= \left(\begin{smallmatrix} a_{11} & a_{12} & a_{13}(z+\varepsilon) & a_{14}(z+\varepsilon)\\ a_{21} & a_{22} & a_{23}(z+\varepsilon) & a_{24}(z+\varepsilon)\\ \frac{a_{31}}{z+\varepsilon} & \frac{a_{32}}{z+\varepsilon} & a_{33} & a_{34}\\ \frac{a_{41}}{z+\varepsilon} & \frac{a_{42}}{z+\varepsilon} & a_{43} & a_{44} \end{smallmatrix}\right),\; A_3= \left(\begin{smallmatrix} a_{11} & a_{12} & a_{13} & a_{14}(z+\varepsilon)\\ a_{21} & a_{22} & a_{23} & a_{24}(z+\varepsilon)\\ a_{31} & a_{32} & a_{33} & a_{34}(z+\varepsilon)\\ \frac{a_{41}}{z+\varepsilon} & \frac{a_{42}}{z+\varepsilon} & \frac{a_{43}}{z+\varepsilon} & a_{44} \end{smallmatrix}\right).\]

By definition, the endomorphism algebra is a subalgebra of \(\bigoplus_{b\in\Delta_0}\mathfrak{gl}_n({\mathbb{K}})\). Its \(z\)-non-negative part – the intersection with the \(\bigoplus_{b\in\Delta_0}\mathfrak{gl}_n({\mathbb{O}})\) – is a free \({\mathbb{O}}\) module generated by the diagonal part \[\label{eq:diag} (E_{i,i},\dots,E_{i,i}),\;1\le i\le n,\tag{5}\] the upper-triangular part \[\label{eq:uppertr} (\underbrace{E_{i,j},\dots,E_{i,j}}_i, \underbrace{(z+\varepsilon)E_{i,j},\dots,(z+\varepsilon)E_{i,j}}_{j-i}, \underbrace{E_{i,j},\dots,E_{i,j}}_{n-j})\tag{6}\] for all \(1\le i <j \le n\), and the lower-triangular part \[\label{eq:lowertr} (\underbrace{(z+\varepsilon)E_{j,i},\dots,(z+\varepsilon)E_{j,i}}_i, \underbrace{E_{j,i},\dots,E_{j,i}}_{j-i}, \underbrace{(z+\varepsilon)E_{j,i},\dots,(z+\varepsilon)E_{j,i}}_{n-j})\tag{7}\] for all \(1\le i <j \le n\). We denote the Lie algebra generated by the elements above by \({\mathfrak a}(\varepsilon)\).

Let \(D\) be a cyclic graded finite-dimensional \(\mathfrak{sl}_n({\mathbb{O}})\)-module with cyclic vector \(v\) (one can replace \(\mathfrak{sl}_n({\mathbb{O}})\) with \(\mathfrak{sl}_n[z]\)). We assume that

• \(v\) is a weight vectors, \(hv=\lambda(h)v\), \(h\in\mathfrak{h}\) for some weight \(\lambda\in \mathfrak{h}^*\),

• \({\mathfrak n}_-[z] v=0\) and \(z\mathfrak{h}[z]v=0\),

• \(\mathrm U({\mathfrak n}[z])v=D\).

In particular, \(D\) is cyclic as the Iwahori algebra module. Recall the Chevalley generators \(e_b\in{\mathfrak n}\), \(e_b=E_{b,b+1}\). For a complex number \(\varepsilon\ne 0\) and \(b=1,\dots,n-1\) let \(\mathrm{sh}_b:{\mathfrak n}({\mathbb{O}})\to {\mathfrak n}({\mathbb{O}})\) be a Lie algebra endomorphism defined by \[\label{eq:sheb} e_bz^k \mapsto e_bz^k(z+\varepsilon),\qquad e_{a}z^k \mapsto e_az^k, a\ne b.\tag{8}\] Explicitly, for \(1\le i<j\le n\) and \(k\ge 0\) one has: \[\label{eq:sh} \mathrm{sh}_b (E_{i,j}z^k)=\begin{cases} \varepsilon E_{i,j}z^k+ E_{i,j}z^{k+1}, & i\le b<j,\\ E_{i,j}z^k, & \text{ otherwise}. \end{cases}\tag{9}\] One easily checks that \(\mathrm{sh}_b\) is indeed compatible with the Lie bracket.

In other words, \(D\) is isomorphic to the \({\mathfrak n}({\mathbb{O}})\) module obtained from \(D\) by shifting the action using the endomorphism \(\mathrm{sh}_b\).

Now let \(D_b\), \(b\in\Delta_0={\mathbb{Z}}/n{\mathbb{Z}}\) be a collection of \(b\)-admissible \(\mathfrak{sl}_n({\mathbb{O}})\) modules with cyclic vectors \(v_b\) (for \(b=0\) the \(b\)-admissibility condition is empty). We consider the Cartan component in the tensor product of all modules \(D_b\), namely \[\odot_{b\in\Delta_0} D_b = \mathrm U(\mathfrak{sl}_n({\mathbb{O}})).\otimes_b v_b \subset \bigotimes_{b\in\Delta_0} D_b.\] The tensor product \(\bigotimes_{b} D_b\) is acted upon by the algebra \({\mathfrak a}(\varepsilon)\) via the embedding \({\mathfrak a}(\varepsilon)\to \bigoplus_b \mathfrak{gl}_n({\mathbb{O}})\). For \(\varepsilon\ne 0\) we define \[\label{eq:Dveps} {\boldsymbol{D}}(\varepsilon) = \mathrm U({\mathfrak a}(\varepsilon)).\otimes_{b\in\Delta_0} v_b\subset \bigotimes_{b\in\Delta_0} D_b.\tag{10}\]

We define the subspace \[{\boldsymbol{D}}(0)=\lim_{\varepsilon\to 0} {\boldsymbol{D}}(\varepsilon)\subset\bigotimes_{b\in\Delta_0} D_b,\] which is a degeneration of the Cartan component \(\odot_{b} D_b\). Explicitly, the procedure works as follows. Let us consider \(\varepsilon\) as an auxiliary variable (parameter). Then \({\boldsymbol{D}}(\varepsilon)\) sits inside \(\left(\bigotimes_{b\in\Delta_0} D_b\right)[\varepsilon]\) 10 . Let us consider the decreasing filtration of the ambient space by the subspaces \(\varepsilon^r\left(\bigotimes_{b} D_b\right)[\varepsilon]\), \(r\ge 0\). One gets an induced decreasing filtration on \({\boldsymbol{D}}(\varepsilon)\) (the intersection of \({\boldsymbol{D}}(\varepsilon)\) with the filtration in the ambient space) \[{\boldsymbol{D}}(\varepsilon)={\boldsymbol{D}}(\varepsilon)_0 \supset {\boldsymbol{D}}(\varepsilon)_1\supset \dots.\] In particular, for each \(r\ge 0\) one gets a map \[\label{eq:vepsto0} {\boldsymbol{D}}(\varepsilon)_r/{\boldsymbol{D}}(\varepsilon)_{r+1} \to \bigotimes_{b\in\Delta_0} D_b,\; p\mapsto \frac{p}{\varepsilon^r}|_{\varepsilon=0},\tag{11}\] \(p\in\left(\bigotimes_{b\in\Delta_0} D_b\right)[\varepsilon]\). Then \({\boldsymbol{D}}(0)\) is spanned by the images of all the maps 11 . In particular, one has the following lemma.

The goal of the rest of the paper is to describe the \({\mathfrak I}\) module \({\boldsymbol{D}}(0)\) in certain special cases.

6 Finite-dimensional representations↩︎

In this section we consider the simplest class of representations of the current algebras: the finite-dimensional representations of \(\mathfrak{sl}_n\). Even in this very special case the general construction produces non-trivial representations of the Iwahori algebra. We start with introducing the one-parameter family of Lie algebras (a shadow of the general picture) and then describe the corresponding representation theory.

6.1 Lie algebras↩︎

As above, let \(\Delta\) be the equioriented quiver with the set of vertices \(\Delta_0={\mathbb{Z}}/n{\mathbb{Z}}\) and the set of arrows \(Q_1=\{b\to b+1, b\in \Delta_0\}\). Let \(M(\varepsilon)=(M_i)_{i\in \Delta_0}\) be a \(\Delta\)-module of dimension \((n,\dots,n)\) defined as follows. We identify all spaces \(M_i\) with a vector space \(\mathrm{span}\{w_j\}_{j=1}^n\). Let all the maps \(M_{b\to b+1}\) be defined by \(w_j\mapsto w_{j}\) for \(j\ne b+1\) and \(w_{b+1}\mapsto \varepsilon w_{b+1}\) (this is the \(z=0\) specialization of 4 ). In particular, the rank of the composition of \(s\) consecutive maps in \(M(0)\) is equal to \(n-s\).

We denote the endomorphism algebra \({\mathop{\rm End}}_\Delta(M(\varepsilon))\) of \(M(\varepsilon)\) by \({\mathfrak I}_1(\varepsilon)\).

6.2 Fundamental representations↩︎

From Lemma 15 one gets the following explicit description: \({\mathfrak I}={\mathfrak I}_1(0)\simeq {\mathfrak b}\oplus({\mathfrak n}_-)^a\), where \({\mathfrak b}\) is the upper-triangular Borel subalgebra in \(\mathfrak{gl}_n\) and \(({\mathfrak n}_-)^a\) is the abelian Lie algebra with the underlying vector space being the space of strictly lower triangular matrices (see [16], [19], [20]). The space \(({\mathfrak n}_-)^a\) is an abelian ideal in \({\mathfrak I}_1\) and the action of \({\mathfrak b}\) on the ideal is induced by the isomorphism \(({\mathfrak n}_-)^a\simeq \mathfrak{gl}_n/{\mathfrak b}\).

Recall the embedding \({\mathfrak I}_1\subset \bigoplus_{b} \mathfrak{gl}_n\). The projection to the \(b\)-th summand \({\mathfrak I}_1\to\mathfrak{gl}_n\) produces the fundamental representations \(V_{\omega _k}^{(b)}\) of \({\mathfrak I}_1\) for all \(k=1,\dots,n-1\) and \(b=0,\dots,n-1\). In particular, for all \(b\) one has the vector space isomorphism \(V_{\omega _k}^{(b)}\simeq \Lambda^k({\mathbb{C}}^n)\).

Recall the basis \(w_1,\dots,w_n\) of the \(n\)-dimensional ambient vector space \({\mathbb{C}}^n\).

6.3 Family of modules↩︎

Let \(\lambda=\sum_{i=1}^{n-1} m_i\omega _i\) be a dominant integral \(\mathfrak{sl}_n\) weight. We fix a decomposition \(\lambda=\sum_{b\in \Delta_0} \lambda^{(b)}\) into a sum of \(n\) dominant integral weights (one for each vertex of our cyclic quiver). In what follows we denote by \(\bar\lambda\) the collection \((\lambda^{(b)})_{b\in \Delta_0}\). Each dominant integral weight \(\lambda\) gives rise to an irreducible highest weight \(\mathfrak{sl}_n\) module \(V_{\lambda}\) with lowest weight vector \(v_\lambda\); we naturally extend the \(\mathfrak{sl}_n\) action to the \(\mathfrak{gl}_n\) action (considering \(\lambda\) as a partition with the last zero entry).

The Cartan embedding \(V_\lambda\hookrightarrow \bigotimes_{b\in \Delta_0} V_{\lambda^{(b)}}\), \(v_\lambda\mapsto \otimes_{b\in \Delta_0} v_{\lambda^{(b)}}\) realizes \(V_\lambda\) inside the tensor product; by definition, \(V_\lambda=\mathrm U(\mathfrak{sl}_n)\cdot \otimes_{b\in \Delta_0} v_{\lambda^{(b)}}=\bigodot_b V_{\lambda^{(b)}}\). The tensor product admits a natural structure of \({\mathfrak I}_1(\varepsilon)\)-module for any \(\varepsilon\) induced via the embedding \({\mathfrak I}_1(\varepsilon)\hookrightarrow \bigoplus_{b\in \Delta_0} \mathfrak{gl}_n\). For \(\varepsilon\ne 0\) we define \[V_{\bar\lambda}(\varepsilon)=\mathrm U({\mathfrak I}_1(\varepsilon))\cdot \otimes_{b\in \Delta_0} v_{\lambda^{(b)}}\subset \bigotimes_{b\in \Delta_0} V_{\lambda^{(b)}}.\]

For \(\varepsilon=0\) we set \[V_{\bar\lambda}(0)=\lim_{\varepsilon\to 0} V_\lambda(\varepsilon) \subset \bigotimes_{b\in \Delta_0} V_{\lambda^{(b)}}.\] The spaces \(V_{\bar\lambda}=V_{\bar\lambda}(0)\) form a special case of the modules \({\boldsymbol{D}}(0)\) from the previous section.

6.4 Algebraic coherence conjecture↩︎

Recall the Weyl group \(W=S_n\). For a dominant integral weight \(\lambda\) and \(\sigma\in W\) let \(v_{\sigma\lambda}\in V_\lambda\) be an extremal weight vector of weight \(\sigma\lambda\). In particular, if \(\sigma=e\), then \(v_{\sigma\lambda}\) is a highest weight vector and if \(\sigma=w_0\) is the longest element, then \(v_{\sigma\lambda}=v_\lambda\) is the lowest weight vector. Let \(V_{\bar\lambda}=V_{\bar\lambda}(0)\).

Lemmas 23 and 24 imply that \(V_{\bar\lambda}\supset \sum_{\sigma\in W} \mathrm U({\mathfrak I}_1) v_{\sigma{\bar\lambda}}\). Since \(\dim V_{\bar\lambda}= \dim V_{\sum_b \lambda^{(b)}}\), Conjecture 25 is equivalent to the equality \[\label{eq:numconj} \dim \sum_{\sigma\in W} \mathrm U({\mathfrak I}_1) v_{\sigma{\bar\lambda}} = \dim V_{\sum_b \lambda^{(b)}}.\tag{12}\]

6.5 Embeddings↩︎

Let \(\lambda^{(b)}=\sum_{i=1}^{n-1} m^{(b)}_i\omega _i\), \({\bar\lambda}=(\lambda^{(b)})_b\). We consider the collections \({\bar\lambda}(i)\), \(i=1,\dots,n-1\), formed by weights which are multiples of \(\omega _i\). More precisely, \[{\bar\lambda}(i) = (m^{(0)}_i\omega _i,m^{(1)}_i\omega _i,\dots, m^{(n-1)}_i\omega _i).\] Then Theorem 27 tells us that \(V_{{\bar\lambda}(i)}\) is generated from the extremal weight vectors.

7 Demazure modules↩︎

In this section we study the general construction of the modules \({\boldsymbol{D}}(0)\) for the case of spherical affine Demazure modules, generalizing the discussion from the previous section.

Let \(D_b\), \(b=0,\dots,n-1\) be \(\mathfrak{sl}_n\)-invariant affine Demazure modules. Let us fix the lowest weight vectors \(v_b\in D_b\). The subspace generated by the \(\mathfrak{sl}_n\) action on \(v_b\) is isomorphic to the highest weight module \(V_{\lambda^{(b)}}\). For a Weyl group element \(\sigma\in S_n\) let \(v_{\sigma\lambda^{(b)}}\in V_{\lambda^{(b)}}\subset D_b\) be the extremal vector of the weight \(\sigma\lambda^{(b)}\).

In what follows we denote \(\bigotimes_{b\in \Delta_0}v_{\sigma\lambda^{(b)}}\) by \(d_{\sigma}\). In particular, the vector \(d_{w_0}\) is the tensor product of the lowest weight vectors of the representations \(V_{\lambda^{(b)}}\subset D_b\). Our goal is to describe \({\boldsymbol{D}}(0)\) as a module over the Iwahori algebra \({\mathfrak I}\).

7.1 Multiples of a level one weight↩︎

Let us fix a dominant integral weight \(\lambda\) and a number \(k\in{\mathbb{Z}}_{\ge 0}\). Let \(D_\lambda\subset L(\Lambda_j)\) be the Demazure module inside integrable level one irreducible representations of affine \(\mathfrak{sl}_n\) such that the weight of the highest weight vector of \(D_\lambda\) is equal to \(\lambda\). Let \(D_{k,\lambda}\) be the level \(k\) Demazure module defined as \(D_\lambda^{\odot k}\). We fix a composition \({\boldsymbol{k}}=(k_b)_{b=0}^{n-1}\) such that \(|{\boldsymbol{k}}|=\sum_{b=0}^{n-1} k_i=k\) and consider the Demazure modules \(D_b=D_{k_b,\lambda}\). The general construction above defines the space \({\boldsymbol{D}}(0)\) inside \(\bigotimes_{b\in \Delta_0} D_{k_b,\lambda}\). We denote this special fiber \({\boldsymbol{D}}(0)\) by \(D_{{\boldsymbol{k}},\lambda}\).

Let \(\Lambda_{\boldsymbol{k}}=\sum_{b\in\Delta_0} k_b \Lambda_b\); in particular, the level of \(\Lambda_{\boldsymbol{k}}\) is equal to \(k\). Recall the subspaces \(\mathrm U({\mathfrak I}).\bigotimes_{b\in \Delta_0}v_{\sigma\lambda^{(b)}}\) inside \(D_{{\boldsymbol{k}},\lambda}\) from Corollary 33, where the Iwahori algebra acts via the isomorphism \({\mathfrak I}\simeq {\mathfrak a}(0)\) (see Lemma 7); in our case \(\lambda^{(b)}=k_b\lambda\).

7.2 Demazure generalization and Kostant–Kumar modules↩︎

Let us consider a more general class of collections of Demazure modules. Let \(w\) be an element of the affine Weyl group and let \(\overline{\Lambda}=(\Lambda^{(0)},\dots\Lambda^{(n-1)})\) be a collection of affine integral dominant weights. We consider the Demazure modules \(D_w(\Lambda^{(b)})\) and assume that all these Demazure modules are spherical (i.e. \(\mathfrak{sl}_n[z]\)-invariant). The Cartan component inside the tensor product of these Demazure modules is described as \[\bigodot_{b=0}^{n-1} D_w(\Lambda^{(b)})\simeq D_w(\sum_b\Lambda^{(b)}).\] Let \({\boldsymbol{D}}(0)\) be the corresponding degeneration (a module of the Iwahori algebra) of \(D_w(\sum_b\Lambda^{(b)})\). It is natural to ask when does the algebraic coherence conjecture holds true, i.e. when is \({\boldsymbol{D}}(0)\) generated by the action of the Iwahori algebra from the set of extremal weight vectors. We provide some examples and non-examples in section 8.

Now let us explain the connection between the modules \({\boldsymbol{D}}(0)\) as above and the Kostant–Kumar modules. Let us fix an affine Weyl group element \(w\) and the affine weights \(\Lambda^{(b)}\). Let us decompose each weight \(\Lambda^{(b)}=\sum_{j=0}^{n-1} k_b^{(j)}\Lambda_j\) as a linear combination of affine fundamental weights. Then one has an embedding \[\label{eq:tpemb} D_w(\Lambda^{(b)})\subset \bigotimes_{j=0}^{n-1} D_w(\Lambda_j)^{\otimes k_b^{(j)}}.\tag{13}\] Let us define the finite weights \(\lambda(j)\) such that \(D_{\lambda(j)}=D_w(\Lambda_j)\). Then 13 induces the embedding \({\boldsymbol{D}}(0)\subset \bigotimes_{j=0}^{n-1} D_{{\boldsymbol{k}}(j),\lambda(j)}\), where \({\boldsymbol{k}}(j)= (k_b^{(j)})_b\). Now similarly to Corollary 30 (assuming algebraic coherence conjecture holds true), one gets the isomorphism \({\boldsymbol{D}}(0)\simeq \sum_{\sigma\in W} K(\bar w(\sigma),\bar \Lambda)\), where \[\bar w(\sigma) = (z^{\sigma\lambda(0)},\dots,z^{\sigma\lambda(n-1)}),\; \bar \Lambda = (\sum_{b=0}^{n-1} k_b^{(0)}\Lambda_b,\dots,\sum_{b=0}^{n-1} k_b^{(n-1)}\Lambda_b).\]

8 Examples↩︎

In this section we provide several explicit low rank examples of the general construction. The irreducible \(\mathfrak{sl}_n\) modules and the affine Demazure modules considered below are generated from the highest weight vectors (as opposed to the lowest weight vectors in the main body of the paper). This choice allows simpler visualization of the examples.

8.1 Rank one case↩︎

We start with the case of \({\mathfrak g}=\mathfrak{sl}_2\), we use the notation from section 6. Let \(e,h,f\) be the standard basis of \(\mathfrak{sl}_2\). For two non-negative integers \(\lambda_0,\lambda_1\ge 0\) we consider the tensor product \(V_{\lambda_0\omega}\otimes V_{\lambda_1\omega}\) of two irreducible representations of \(\mathfrak{sl}_2\) with highest weight vectors \(v_{\lambda_0}\) and \(v_{\lambda_1}\). The Weyl group consists of two elements \(\mathrm{id}\) and \(s\); we denote by \(v_{s\lambda}\in V_ \lambda\) the lowest weight (extremal) vector. The operator \(f(\varepsilon) = f\otimes\mathrm{id} + \varepsilon\mathrm{id}\otimes f\) being applied multiple times to \(v_{\lambda_0}\otimes v_{\lambda_1}\) produces the following vectors \[\begin{gather} f(\varepsilon)^m v_{\lambda_0}\otimes v_{\lambda_1} = \sum_{i=0}^m \varepsilon^{m-i} f^{i}v_{\lambda_0}\otimes f^{m-i}v_{\lambda_1},\;0\le m\le \lambda_0,\\ f(\varepsilon)^{\lambda_0+m} v_{\lambda_0}\otimes v_{\lambda_1} = \varepsilon^m \sum_{i=0}^{\lambda_0} \varepsilon^{\lambda_0-i} f^iv_{\lambda_0}\otimes f^{\lambda_0+m-i}v_{\lambda_1}, 1\le m\le \lambda_1. \end{gather}\] The lowest \(\varepsilon\)-degree parts of the above vectors are equal to \[\begin{gather} \tag{14} (f(\varepsilon)^m v_{\lambda_0}\otimes v_{\lambda_1})^o = f^mv_{\lambda_0}\otimes v_{\lambda_1},\;0\le m\le \lambda_0,\\ \tag{15} (f(\varepsilon)^{\lambda_0+m} v_{\lambda_0}\otimes v_{\lambda_1})^o = f^{\lambda_0}v_{\lambda_0}\otimes f^{m}v_{\lambda_1}, 1\le m\le \lambda_1. \end{gather}\] Vectors 14 and 15 span the space \(V_{\bar\lambda}\). The Lie algebra \({\mathfrak I}_1\subset \mathfrak{gl}_2\oplus\mathfrak{gl}_2\) is the span of three elements \((f,0)\), \((0,e)\) and \((h,h)\). One sees explicitly that \[\begin{gather} \mathrm U({\mathfrak I}_1)(v_{\lambda_0}\otimes v_{\lambda_1}) = \mathrm{span}\{f^mv_{\lambda_0}\otimes v_{\lambda_1},\;0\le m\le \lambda_0\},\\ \mathrm U({\mathfrak I}_1)(v_{s\lambda_0}\otimes v_{s\lambda_1}) = \mathrm{span}\{f^{\lambda_0}v_{\lambda_0}\otimes f^{m}v_{\lambda_1}, 0\le m\le \lambda_1\}, \end{gather}\] and the intersection of these two spaces is the span of the vector \(v_{s\lambda_0}\otimes v_{\lambda_1}\) (we note that \(f^{\lambda_0}v_{\lambda_0}\) is proportional to \(v_{s\lambda_0}\) and \(f^{m}v_{\lambda_1}\) is proportional to \(e^{\lambda_1-m}v_{s\lambda_1}\)). Hence, \(V_{{\bar\lambda}}\) is generated from two vectors \(v_{\lambda_0}\otimes v_{\lambda_1}\) and \(v_{s\lambda_0}\otimes v_{s\lambda_1}\) by the action of the Lie algebra \({\mathfrak I}_1\). We also see that \(V_{\bar\lambda}\) is cocylic with the cocyclic vector \(v_{s\lambda_0}\otimes v_{\lambda_1}\).

Let \(\Lambda_0\) and \(\Lambda_1\) be integrable level one irreducible representations of \(\widehat{\mathfrak{sl}}_2\). There is an embedding \(V_{\bar\lambda}\hookrightarrow L(\Lambda)\) with \(\Lambda= \lambda_0\Lambda_1+\lambda_1\Lambda_0\). Let \(u_\Lambda\in L(\Lambda)\) be the highest weight vector and let \(s_0,s_1\) be the standard generators of the affine Weyl group of type \(A_1^{(1)}\). In particular, \(L(\Lambda)\) contains the extremal weight vectors \(u_{s_0\Lambda}\) and \(u_{s_1\Lambda}\). Then the embedding sends \[v_{s\lambda_0}\otimes v_{\lambda_1}\mapsto u_\Lambda,\; v_{\lambda_0}\otimes v_{\lambda_1}\mapsto u_{s_0\Lambda},\; v_{s\lambda_0}\otimes v_{s\lambda_1}\mapsto u_{s_1\Lambda}.\]

Here is the picture for \(\lambda_0=2\), \(\lambda_1=3\). We start with the tensor product \[\begin{tikzcd}[scale cd=1] \bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] v_2\quad \otimes\quad \bullet & \arrow[l,"f",swap]\bullet &\arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] v_3, \end{tikzcd}\] consider inside the Cartan component \[\begin{tikzcd} \bullet& \arrow[l,"f",swap]\bullet & \arrow[l,"f",swap]\bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] v_2\otimes v_3 \end{tikzcd}\] and then deform it to the Iwahori algebra module \[\begin{tikzcd} & & &\bullet & \arrow[l,"ft^0",swap]\bullet &\arrow[l,"ft^0",swap] \bullet \\ & & \bullet\arrow[ur,"et"] \\ & \bullet\arrow[ur,"et"] \\ \bullet\arrow[ur,"et"] \end{tikzcd}\] which is a subspace in the irreducible integrable \(\widehat{\mathfrak{sl}}_2\) module \(L(3\Lambda_0+2\Lambda_1)\) (this subspace a sum of two Demazure modules corresponding to the length one elements \(s_0\) and \(s_1\) in the affine Weyl group).

We note that the same Cartan component \(V_{5\omega}\) \[\begin{tikzcd} \bullet& \arrow[l,"f",swap]\bullet & \arrow[l,"f",swap]\bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] v_2\otimes v_3 \end{tikzcd}\] sits inside another tensor product \[\begin{tikzcd}[scale cd=1] \bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] \bullet & \arrow[l,"f",swap] v_2\quad \otimes\quad \bullet & \arrow[l,"f",swap] v_3. \end{tikzcd}\] One obtains another degeneration of \(V_{5\omega}\) of the form \[\begin{tikzcd} & \bullet & \arrow[l,swap,"ft^0"] \bullet & \arrow[l,swap,"ft^0"] \bullet & \arrow[l,swap,"ft^0"] \bullet &\arrow[l,swap,"ft^0"] \bullet \\ \bullet \arrow[ur,"et"] \end{tikzcd}\] which is the sum of two Demazure modules inside the irreducible integrable \(\widehat{\mathfrak{sl}}_2\) module \(L(\Lambda_0+4\Lambda_1)\).

8.2 Rank two finite case↩︎

Let us consider \(\mathfrak{sl}_3\) and \(\lambda_0=\omega _1\), \(\lambda_1=\omega _2\), \(\lambda_2=\omega _1\). This is one of the simplest examples which does not show up in the context of the geometric coherence conjecture. The \(15\)-dimensional Cartan component is isomorphic to \(V_{2\omega _1+\omega _2}\); let us describe its degeneration to the Iwahori module \(V_{\bar\lambda}\).

Let \(v_1,v_2,v_3\) and \(v_{12}, v_{13}, v_{23}\) be the standard bases of \(V_{\omega _1}\) and \(V_{\omega _2}\). Then the cocyclic vector in \(V_{\bar\lambda}\) is \(v_3\otimes v_{23}\otimes v_1\). We want to check that the action of \({\mathfrak I}_1\) (the Iwahori quotient) generates \(15\)-dimensional subspace from the six extremal weight vectors inside \(V_{\lambda_0}\otimes V_{\lambda_1}\otimes V_{\lambda_2}\). The six operators we use are as follows: \[\begin{gather} f_1t^0 = (f_1,f_1,0),\;f_2t^0 = (f_2,0,f_2),\;f_{12}t^0=(f_{12},0,0),\\ e_1t = (0,0,e_1),\;e_2t = (0,e_2,0),\; e_{12}t = (0,e_{12},e_{12}). \end{gather}\] Here is the picture for \(V_{\bar\lambda}\): \[\begin{tikzcd}[scale cd=0.7] 0 & & & \underline{v_1v_{12}v_1} \arrow[dl,"f_1"description] & \\ 0 & & v_2v_{12}v_1 \arrow[dr,"f_2"description]& & \\ 0 & & & \boxed{v_3v_{12}v_1} & \\ 1 & \underline{v_2v_{12}v_2} \arrow[uur,"e_1t"description] \arrow[dr,"f_2"description] & & v_2v_{13}v_1 + v_1v_{23}v_1 \arrow[ddl]& \underline{v_1v_{13}v_1} \arrow[uuul,"e_2t"description]\arrow[l]\arrow[d,"f_{12}"description]\\ 1 & & v_3v_{12}v_2 + v_2v_{12}v_3\arrow[r] & v_3v_{12}v_3 \arrow[uu,bend left=70] & v_3v_{13}v_1\arrow[uul]\\ 1 & & v_2v_{23}v_1 \arrow[r] & v_3v_{23}v_1 \arrow[uuu,bend left=80] & \\ 2 & \underline{v_2v_{23}v_2}\arrow[ur,"e_2t"description]\arrow[uuu,"e_{12}t"description] \arrow[dr,"f_2"description]& & & \underline{v_3v_{13}v_3}\arrow[ddl,"f_1"description]\arrow[uu,"e_{12}t"description]\arrow[uul]\\ 2 & & v_3v_{23}v_2 + v_2v_{23}v_3 \arrow[dr,"f_2"description]& & \\ 2 & & & \underline{v_3v_{23}v_3} \arrow[uuu,"e_{12}t"]\arrow[uuuu] &\\ & \end{tikzcd}\] On the picture the extremal weight vectors are underlined, the cocyclic vector is boxed and the numbers on the left correspond to the standard \(z\)-degree of the Iwahori algebra (we note that the arrow from the lowest weight vector \(v_3v_{23}v_3\) points to two different vectors, which means that the result of the application of the operator \(e_{12}z\) is a linear combination of the vectors \(v_3v_{23}v_1\) and \(v_3v_{12}v_3\)).

8.3 Rank two affine case↩︎

In this subsection we provide several examples for the construction from section 7.2 for the \(\widehat{\mathfrak{sl}}_2\) Demazure modules.

Let \(\Lambda^{(0)}=\Lambda^{(1)}=\Lambda_0+\Lambda_1\) and let \(w=s_1s_0\). The spherical Demazure module \(D_{s_1s_0}(\Lambda^{(0)}+\Lambda^{(1)})\) sits inside \(D_{s_1s_0}(\Lambda^{(0)})\otimes D_{s_1s_0}(\Lambda^{(1)})\); it is of dimension \(15\) and can be visualized as follows:

\[\label{eq:3425} \begin{tikzcd}[scale cd=0.5] & & \bullet &\bullet &\bullet \\ & \bullet &\bullet &\bullet &\bullet &\bullet \\ \bullet & \bullet &\bullet &\bullet &\bullet &\bullet &\bullet \\ & \end{tikzcd}\tag{16}\] The dots in the picture correspond to the basis vectors, vectors of the same \(\mathfrak{h}\)-weight are placed in one column, vectors of the same z-degree are placed in the same rows. The tensor product \(D_{s_1s_0}(\Lambda^{(0)})\otimes D_{s_1s_0}(\Lambda^{(1)})\) is visualized as

\[\begin{tikzcd}[scale cd=0.5] & \bullet &\bullet \\ \bullet &\bullet &\bullet &\bullet\\ & \end{tikzcd} \qquad \otimes\qquad \begin{tikzcd}[scale cd=0.5] & \bullet &\bullet \\ \bullet &\bullet &\bullet &\bullet \\ & \end{tikzcd}\]

The Iwahori algebra module \({\boldsymbol{D}}(0)\) is generated from two extremal vectors (the tensor squares of highest and lowest weight vector) and has the following form

\[\label{eq:3425Iw} \begin{tikzcd}[scale cd=0.4] & & &\bullet & _\bullet^\bullet &\bullet & \\ & & _\bullet^\bullet &_\bullet^\bullet &\bullet & \bullet & \bullet \\ & \bullet & \bullet & & & & &\\ & \bullet & & & & &\\ \bullet & & & & & & \end{tikzcd}\tag{17}\] (the doubled points indicate that the corresponding weight-degree space is two-dimensional). In particular, \(\dim{\boldsymbol{D}}(0)=\dim D_{s_1s_0}(\Lambda^{(0)}+\Lambda^{(1)})=15\) and the characters coincide as well (compare 16 and 17 ). We note that this example does not show up in the geometric coherence conjecture, since \(\Lambda=\Lambda_0+\Lambda_1\) is not a multiple of an affine fundamental weight.

Now let \(\Lambda^{(0)}=\Lambda_0\), \(\Lambda^{(1)}=\Lambda_1\) and let \(w=s_1s_0s_1\). The spherical Demazure module \(D_{s_1s_0s_1}(\Lambda^{(0)}+\Lambda^{(1)})\) sits inside \(D_{s_1s_0s_1}(\Lambda_0)\otimes D_{s_1s_0s_1}(\Lambda_1)\); it is of dimension \(18\) and can be visualized as follows:

\[\label{eq:2423423} \begin{tikzcd}[scale cd=0.5] & & \bullet &\bullet \\ & \bullet &_\bullet^\bullet &_\bullet^\bullet &\bullet \\ & \bullet &\bullet &\bullet &\bullet \\ \bullet & \bullet &\bullet &\bullet &\bullet &\bullet & \end{tikzcd}\tag{18}\] The tensor product \(D_{s_1s_0s_1}(\Lambda_0)\otimes D_{s_1s_0s_1}(\Lambda_1)\) is visualized as

\[\begin{tikzcd}[scale cd=0.4] & \bullet \\ \bullet &\bullet &\bullet \\ & \end{tikzcd} \qquad \otimes\qquad \begin{tikzcd}[scale cd=0.5] & \bullet &\bullet \\ & \bullet &\bullet \\ \bullet &\bullet &\bullet &\bullet\\ & \end{tikzcd}\]

The Iwahori algebra module \({\boldsymbol{D}}(0)\) is generated from two extremal vectors (the tensor squares of highest and lowest weight vector) and has the following form

\[\label{eq:2423423Iw} \begin{tikzcd}[scale cd=0.4] & & &\bullet \\ & & _\bullet^\bullet &_\bullet^\bullet &_\bullet^\bullet \\ & \bullet & _\bullet^\bullet & _\bullet^\bullet & \bullet & \bullet \\ & \bullet &\bullet \\ & \bullet \\ \bullet \end{tikzcd}\tag{19}\] One has \(\dim{\boldsymbol{D}}(0)=\dim D_{s_1s_0s_1}(\Lambda^{(0)}+\Lambda^{(1)})=18\) and the characters coincide as well (compare 18 and 19 ).

8.4 Rank two affine: non-example↩︎

We start with the Demazure module \(D_{s_1s_0}(2\Lambda)\), \(\Lambda=\Lambda_0+\Lambda_1\) 16 , but now embed \(D_{s_1s_0}(2\Lambda)\) into the tensor product \(D_{s_1s_0}(\Lambda_0)\otimes D_{s_1s_0}(\Lambda_0+2\Lambda_1)\) as a Cartan component. This tensor product is visualized as \[\begin{tikzcd}[scale cd=0.5] &\bullet & \\ \bullet &\bullet &\bullet \\ & \end{tikzcd} \qquad \otimes\qquad \begin{tikzcd}[scale cd=0.5] & \bullet &\bullet &\bullet \\ \bullet &\bullet &\bullet &\bullet &\bullet \\ & \end{tikzcd}\] Then \(\dim{\boldsymbol{D}}(0)=\dim D_{s_1s_0}(2\Lambda)=15\), but the action of the Iwahori algebra \({\mathfrak I}\) generates from the two extremal vector the \(14\)-dimensional representation which is visualized as follows:

\[\begin{tikzcd}[scale cd=0.5] & &\bullet & _\bullet^\bullet &\bullet & \bullet & \\ & \bullet & _\bullet^\bullet & \bullet &\bullet & \bullet & \bullet \\ & \bullet & & & & & \\ \bullet & & & & & & \end{tikzcd}\]

9 Program↩︎

In the appendix we describe the computer program which verifies Conjecture 25 by comparing the dimension of \(\sum_{\sigma\in W} \mathrm U({\mathfrak I}_1) v_{\sigma{\bar\lambda}}\) with the dimension of \(V_{\sum_b \lambda^{(b)}}\). In all the performed checks the dimensions do coincide.

9.1 The setup↩︎

Let \(\lambda=(\lambda_1\ge\dots\ge\lambda_n\ge 0)\) be a partition, \(V_\lambda\) the corresponding irreducible \(\mathfrak{gl}_n\) module with the set of extremal weight vectors \(v_{\sigma\lambda}\in V_\lambda\), \(\sigma\in S_n\). The representation \(V_\lambda\) admits the Gelfand-Tsetlin basis labeled by the Gelfand-Tsetlin diagrams (GT patterns) \(\underline\lambda=(\lambda_{k,i})\) (see [37]); we denote the vector corresponding to \(\underline\lambda\) by \(v(\underline\lambda)\in V_\lambda\). The GT pattern \(\underline\lambda(\sigma)\) corresponding to the extremal vector \(v_{\sigma\lambda}\) is described as follows: the \(k\)-th line (the numbers \(\lambda_{k,i}\), \(1\le i\le k\)) are the numbers \(\lambda_{\sigma(1)}, \lambda_{\sigma(2)},\dots, \lambda_{\sigma(k)}\) placed in the non-decreasing order.

Our input is a collection of \(n\) partitions \(\lambda^{(b)}\), \(b\in{\mathbb{Z}}/n{\mathbb{Z}}\) labeled by the vertices of our cyclic quiver, \(\lambda^{(b)}=(\lambda^{(b)}_1\ge\dots\ge \lambda^{(b)}_n)\). We are working inside the tensor product \(V=\bigotimes_{b=0}^{n-1} V_{\lambda^{(b)}}\) of the corresponding irreducible representations. In what follows we denote by \(\underline\lambda^{(b)}\) the GT patterns corresponding to the weight \(\lambda^{(b)}\). We have a basis of \(V\) formed by the tensor products \(\otimes_{b=0}^{n-1} v(\underline\lambda^{(b)})\), where \(\underline\lambda^{(b)}\) are arbitrary GT patterns corresponding to \(\lambda^{(b)}\).

As an output the program computes the dimension of a subspace of \(V\) generated by the action of \({\mathfrak I}_1\) from \(n!\) extremal weight vectors \[w(\sigma)=\bigotimes_{b=0}^{n-1} v_{\sigma\lambda^{(b)}},\;\sigma\in S_n.\] The Lie algebra \({\mathfrak I}_1\) is generated by \(n\) operators acting on our tensor product \(V\); let us denote these operators by \(F_b\), \(b=0,\dots,n-1\). Explicitly, for \(b=1,\dots,n-1\) one has \(F_b=\sum_{\substack{0\le a\le n-1\\ a\ne b}} E^{(a)}_{b+1,b}\) and \(F_0=\sum_{a=1}^{n-1} E_{1,n}^{(a)}\) (for \(x\in\mathfrak{gl}_n\) the operators \(x^{(a)}\) acts as \(x\) on the \(a\)-th factor and as identity on other factors).

The tensor product \(V\) is a representation of \({\mathfrak I}_1\) and we are interested in the dimension of the subspace \(S=\mathrm{U}({\mathfrak I}_1).\mathrm{span}\bigl\{w(\sigma),\;\sigma\in S_n\bigr\}\). Our goal is to check that \(S=V_{\bar\lambda}\) (see Section 6), which is equivalent to \(\dim S = \dim V_{\sum_b \lambda^{(b)}}\).

9.2 The algorithm↩︎

At each stage we have a list of linearly independent vectors from \(S\); each vector from the list is a weight vector. In particular, the weight of the extremal vector \(v_{\sigma\lambda}\in V_\lambda\) is equal to \(\sigma(\lambda)\). As a consequence, the weight of the vector \(w(\sigma)\in \bigotimes_{a=0}^{n-1} V_{\lambda^{(a)}}\) is equal to \(\sum_{a=0}^{n-1} \sigma(\lambda^{(a)})\).

Given a linearly independent generating set \(\mathcal{S}\) (e.g. \(\{w(\sigma):\sigma \in S_n\}\)) the program generates \(S\) as follows: First, it defines the set of all vectors to which the operators \(F_a\) have not yet been applied, setting \(V:=\mathcal{S}\). Then, the program proceeds by taking out an element \(v\in V\) at each iteration and calculating \(F_a v\). Non-zero results are added to the temporary basis of \(S\) only if they are linearly independent of the elements of \(S\) having the same weight. Such results are also added to \(V\) for being used in the next iterations. The iteration continue this way until there are no more vectors to apply the operators on (i.e. \(V=\emptyset\)).

9.3 Pseudocode↩︎

We now present the algorithm described in 9.2 as pseudocode. ¹. The function that computes \(\dim S\) (and in fact generates a basis for \(S\)) is called generate_subspace and can be found in main.py. Also, for for any vector \(u\) let \(\mu(u)\) be the weight of \(u\). The code works as follows:

References↩︎

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1. The full Python code can be found here: https://github.com/AndreyK19/quiver_subspace↩︎