On the structure of Weyl modules

Introduction Let G be a simply connected and connected semisimple group over an algebraically closed field k of characteristic p > 0. T ⊂ G is a maximal torus and R is the root system relative to T . X(T ) is the weight lattice. Let B ⊃ T be a Borel subgroup corresponding to the negative roots R− = R. Denote by Gr the r-th Frobenius kernel of G. The socle and radical structures of the cohomology groups of line bundles on the flag variety G/B are determined by those structures of the GrT -modules Zr(λ) = Ind GrT BrTλ (cf. [12]). So the study of the GrT -structure of these modules turns out to be more interesting. Calculation of the extensions between simple modules plays an important rule in determining the socle structure. In this paper, we calculate the socle series of the Weyl modules with p-singular highest weight for the group of type G2 by studying the extensions between simple modules for Frobenius kernel. In the first section of this paper, we study the properties of Ext Gr(L(μ), H (λ)), which turns out to be semisimple and to have a good filtration for large p and prestricted weights μ and λ. Some of the vanishing properties of these modules are also studied. Then we use these properties to calculate Ext Gr(L(μ), L(λ)), which will lead a calculation of the extensions between simple GrT -modules. The results in this section will be used in Section 3 to calculate the socle series of Z1(λ) with p-singular weights λ for the group of type G2. The author ([13] ch3) used the method of Doty and Sullivan [7] and calculated the socle series of H(λ) for the p-singular weights λ in the bottom p-alcove for the groups of type A2 and B2. However when the multiplicities of simple modules in

Higher Obstructions to Deforming Cohomology Groups of Line Bundles

Let $$\mathfrak {g}$$ be a symmetrizable Kac-Moody Lie algebra with the standard Cartan subalgebra $$\mathfrak {h}$$ and the Weyl group $$W$$ . Let $$P_+$$ be the set of dominant integral weights. For $$\lambda \in P_+$$ , let $$L(\lambda )$$ be the integrable, highest weight (irreducible) representation of $$\mathfrak {g}$$ with highest weight $$\lambda $$ . For a positive integer $$s$$ , define the saturated tensor semigroup as $$\begin{aligned} \Gamma _s:= \{(\lambda _1, \dots , \lambda _s,\mu )\in P_+^{s+1}: \exists \, N\ge 1 \,\text {with}\,L(N\mu )\subset L(N\lambda _1)\otimes \dots \otimes L(N\lambda _s)\}. \end{aligned}$$ The aim of this paper is to begin a systematic study of $$\Gamma _s$$ in the infinite dimensional symmetrizable Kac-Moody case. In this paper, we produce a set of necessary inequalities satisfied by $$\Gamma _s$$ . These inequalities are indexed by products in $$H^*(G^{\mathrm{min }}/B; \mathbb {Z})$$ for $$B$$ the standard Borel subgroup, where $$G^{\mathrm{min }}$$ is the ‘minimal’ Kac-Moody group with Lie algebra $$\mathfrak {g}$$ . The proof relies on the Kac-Moody analogue of the Borel-Weil theorem and Geometric Invariant Theory (specifically the Hilbert-Mumford index). In the case that $$\mathfrak {g}$$ is affine of rank 2, we show that these inequalities are necessary and sufficient. We further prove that any integer $$d>0$$ is a saturation factor for $$A^{(1)}_1$$ and 4 is a saturation factor for $$A^{(2)}_2$$ .

Let Γ be a connected reductive group over an algebraically closed field F of positive characteristic and let G be the set of fixed points of Γ under some Frobenius map of Γ. Then G is a finite group of Lie type. The p-modular representations of G for p = char F (the “describing” characteristic case) are closely related to rational representations of Γ, and thus results from the theory of reductive algebraic groups can be used to develop the representation theory of finite groups of Lie type in the describing characteristic case. One of the outstanding open questions in this area is to determine the dimensions of the irreducible representations and the decomposition matrix whose entries record multiplicities of irreducible modules as composition factors of Weyl modules. One way to define Weyl modules is by reducing modulo p (p = char F) the irreducible modules of the complex algebraic group of the same type. Here reduction modulo p happens in both, in the field underlying the group and in the field over which the representing matrices are defined, simultaneously. One of the main tools to study these questions is derived from the Weyl group of Γ. If, for example, Γ is a general linear group, the decomposition matrix of the Weyl modules can be calculated from decomposition numbers of Schur algebras, which in turn can be calculated in terms of symmetric groups. We shall discuss this phenomenon in more detail below.

These notes give the details of a result stated inaprevious paper ([2], Section 7(b)) which asserts that the Knapp-Stein intertwining operator [3] linking the principal series representations of SL(2,~) coming from characters of a split Borel subgroup differing by a Weyl reflection can be constructed from the so-called half-density pairin$ ([2], p. 152), due to Kostant and Sternberg, of geometric quantization. Section 2 of the present paper recalls a construction of [4], which manufactures a hermitian line bundle with connection (L,V) given an orbit X of the eoadjoint representation of a simply connected Lie group G and a point p e X. We also derive some results relating nonvanishing sections of L and certain 1-forms they define. The principal result here is Proposition 2.6. Section 3 is devoted to working out our example. G is the universal covering group of SL(2,~) and X is a hyperboloid of one sheet. Whereas Knapp and Stein fix a Borel subgroup of G and let the Weyl group act on the inducing representation, the geometry of our situation leads us to use the Weyl group to move the Borel subgroup while holding the inducing representation fixed. Section 4 comments on our result and raises questions that need to be settled concerning the half-density pairing. In what follows we shall assume the reader to be familiar with Kostant's fundamental paper [4] on geometric pre-quantization and also with the half-density pairing as set forth in [2].

We build the wrapped Fukaya category W(E) for any monotone symplectic manifold, convex at infinity. We define the open-closed and closed open-string maps. We study their algebraic properties and prove that the string maps are compatible with the eigenvalue splitting of W(E). We extend Abouzaid's generation criterion from the exact to the monotone setting. We construct an acceleration functor from the compact Fukaya category which on Hochschild (co)homology commutes with the string maps and the canonical map from quantum cohomology QH(E) to symplectic cohomology SH(E). We define the QH(E)- and SH(E)-module structure on the Hochschild (co)homology of W(E) which is compatible with the string maps. The module and unital algebra structures, and the generation criterion, also hold for the compact Fukaya category F(E), and also hold for closed monotone symplectic manifolds. As an application, we show that the wrapped category of any monotone negative line bundle over any projective space is proper (cohomologically finite). For any monotone negative line bundle E over a toric Fano variety, we show that SH(E) is non-trivial and that W(E) contains an essential non-displaceable monotone Lagrangian torus.

Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions

Let \(\mathfrak{g}_{0}\) be a real semisimple Lie algebra. Let \(\mathfrak{g}_{0} = \mathfrak{k}_{0} \oplus \mathfrak{p}_{0}\) be the corresponding Cartan decomposition and \(\mathfrak{h}_{0} = \mathfrak{t}_{0} \oplus \mathfrak{a}_{0}\) be a maximally compact Cartan subalgebra of \(\mathfrak{g}_{0}\). Let \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) and \(\mathfrak{h} = \mathfrak{t} \oplus \mathfrak{a}\) be the corresponding complexifications. The set \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) consists of all the linear forms on \(\mathfrak{t}\) which are the restriction to \(\mathfrak{t}\) of the roots in the root system \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\) of \(\mathfrak{g}\) with respect to \(\mathfrak{h}\). The main result of the paper is to prove that \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) is also a (maybe non-reduced) root system and its Weyl group can be identified with a subgroup of the Weyl group of \(\Delta {\left( {\mathfrak{g},\mathfrak{h}} \right)}\). Let \(Spin\,\nu :\mathfrak{k} \to End\,S\) be the composition of the isotropy representation \(\nu :\mathfrak{k} \to \mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)}\) with the spin representation \(Spin:\mathfrak{s}\mathfrak{o}{\left( \mathfrak{p} \right)} \to End\,S\). Finally as an application, we give a nice description of the \(\mathfrak{k}\)-module structure on S in terms of the restricted root system \(\Delta {\left( {\mathfrak{g},\mathfrak{t}} \right)}\) and its Weyl group.

For G a reductive group and $$T\subset B$$ a maximal torus and Borel subgroup, Demazure modules are certain B-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of G. In order to describe the T-weight spaces that appear in a Demazure module, we study the convex hull of these weights—the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that G is simple of classical Lie type. Specializing to $$G=GL_n$$ , we recover results of Fink, Mészáros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.

We prove the multiplicity one case of Lusztig's conjecture on the irreducible characters of reductive algebraic groups for all fields with characteristic above the Coxeter number

Our goal is reached as follows. Fix a maximal torus T in G and a Borel subgroup B ⊃ T , so that we can consider the corresponding root lattice Q, weight lattice Π , set of simple roots ∆ = {α1, . . . ,αn} and set of fundamental weights Ω = {ω1, . . . ,ωn}. We then choose P ⊃ B so that P corresponds to a subset S of ∆ or equivalently Ω or also equivalently of the Dynkin diagram of G. Recall that the Picard group of G/P can be identified with a sublattice of Π . Taken an ample L ∈ Pic(G/P ), and setting λ = ∑

Let G be a reductive algebraic group with a Borel subgroup B. We define the quasi-coherent Hecke category for the pair (G,B). For any regular Noetherian G-scheme X we construct a monoidal action of the Hecke category on the derived category of B-equivariant quasi-coherent sheaves on X. Using the action we define the Demazure Descent Data on the latter category and prove that the Descent category is equivalent to the derived category of G-equivariant sheaves on X.

Generalized Cartan TypeSLie Algebras in Characteristic Zero

In weighted random pattern testing based on deterministic test patterns, if the number of the deterministic test patterns with a low sampling probability is reduced, the number of the weighted random patterns required to achieve a high fault coverage can be decreased. The weight set that makes the variance of sampling probabilities for deterministic test patterns very small, can reduce the number of the deterministic test patterns with a low sampling probability. In this paper, we present a new weight set generation algorithm which can generate high performance weight sets by reducing the variance of the sampling probabilities. Experimental results on ISCAS-85 benchmark circuits prove the effectiveness of the new weight set generation algorithm.

Reductive subgroups of reductive groups in nonzero characteristic