Peter Littelmann

A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras.

In the representation theory of the group GLn(C), an important tool are the Young tableaux. The irreducible representations are in one-to-one correspondence with the shapes of these tableaux. Let T be the subgroup of diagonal matrices in GLn(C). Then there is a canonical way to assign a weight of T to any Young tableau such that the sum over the weights of all tableaux of a fixed shape is the character CharV of the corresponding GLn(C)-module V . Note that this gives not only a way to compute the character, it gives also a possibility to describe the multiplicity of a weight in the representation: It is the number of different tableaux of the same weight. Eventually, the Littlewood-Richardson rule describes the decomposition of tensor products of GLn(C)modules purely in terms of the combinatoric of these Young tableaux.

Cones, crystals, and patterns

There are two well known combinatorial tools in the representation theory ofSLn, the semi-standard Young tableaux and the Gelfand-Tsetlin patterns. Using the path model and the theory of crystals, we generalize the concept of patterns to arbitrary complex semi-simple algebraic groups.

A Generalization of the Littlewood-Richardson Rule

For the general linear group GZ, the Littlewood-Richardson rule (see [18, 193) gives a method to calculate the decomposition of a tensor product of two irreducible G/,-representations. The aim of this article is to give a generalization of this rule for all simple, simply connected algebraic groups of type A,, B,, C,, D,, G2, and E,. We obtain also partial results for G of type F4, E,, and ES. The restrictions in the last three cases come from the fact that for the formulation of the decomposition rules we need the notion of a standard Young tableau. Such a notion has been developed by Seshadri, Lakshmibai, Musili, and Rajeswari in a series of articles (see [ll, 13, 14, 16]), but not yet for all representations of the last three exceptional groups. The advantage of the notion of a Young tableau developed by Seshadri et al. is that it is independent of the type of the group. For the convenience of the reader not used to this notion we give first a seperate proof for G = SZ, using the classical notion of a standard Young tableau. Of course, for applications the classical notion is much more appropriate. In the Appendix we give a “translation” of the notion of a standard Young tableau in the sense of Seshadri et al. into the classical notion of a Young tableau for G = Sp2,,, and Spin,. In 3.8 we give also such a translation for G = G,. For the other exceptional groups such a translation is not known to the author. Let p be a dominant weight for G and denote by I’@ the corresponding simple G-module. We assume that the character of VP is given by CT eYcr), where the sum has to be taken over all standard Young tableaux F of shape p(p) and v(5) is the weight of the tableau y. Now by the

LS galleries, the path model, and MV cycles

We give an interpretation of the path model of a representation [18] of a complex semisimple algebraic group G in terms of the geometry of its affine Grassmannian. In this setting, the paths are replaced by LS–galleries in the affine Coxeter complex associated to the Weyl group of G. To explain the connection with geometry, consider a Demazure–Hansen–Bott–Samelson desingularization ˆ Σ(λ) of the closure of an orbit G(C[[t]]).λ in the affine Grassmannian. The homology of ˆ Σ(λ) has a basis given by Bia lynicki–Birula cell’s, which are indexed by the T –fixed points in ˆ Σ(λ). Now the points of ˆ Σ(λ) can be identified with galleries of a fixed type in the affine Tits building associated to G, and the T –fixed points correspond in this language to combinatorial galleries of a fixed type in the affine Coxeter complex. We determine those galleries such that the associated cell has a non-empty intersection with G(C[[t]]).λ (identified with an open subset of ˆ Σ(λ)), and we show that the closures of the strata associated to LS-galleries are exactly the MV–cycles [24], which form a basis of the representation V (λ) for the Langland’s dual group G ∨ .

Contracting modules and standard monomial theory for symmetrizable Kac-Moody algebras

Let G be a reductive algebraic group defined over an algebraically closed field k. We fix a Borel subgroup B, and for a dominant weight λ let Lλ be the associated line bundle on the generalized flag variety G/B. In a series of articles, Lakshmibai, Musili and Seshadri initiated a program to construct a basis for the space H0(G/B,Lλ) with some particularly nice geometric properties. The purpose of the program is to extend the Hodge-Young standard monomial theory for the group SL(n) to the case of any semisimple algebraic group and, more generally, to Kac-Moody algebras. We refer to [3], [7], [10], [14] for a survey of the subject and applications. We provide a new approach which completes the program and which avoids the case by case considerations of the earlier articles. In fact, the method works for all symmetrizable Kac-Moody algebras. The most important tools we need in our approach are the combinatorial language of the path model of a representation [11], [12], and quantum groups at a root of unity. Let Uv(g) be the quantum group associated to G at an `-th root of unity v. We use the quantum Frobenius map [15] to “contract” certain Uv(g)-modules so that they become G-modules. The corresponding map between the dual spaces can be seen as a kind of splitting of the power map H(G/B,Lλ) → H(G/B,L`λ), s 7→ s. For simplicity let us assume we are in the simply laced case. Let Vλ be the Weyl module of G of highest weight λ, and let Mλ be the corresponding Weyl module of Uv(g). There is a canonical way to attach a tensor product bπ := bν1 ⊗ . . .⊗ bν` of extremal weight vectors bνj ∈ M∗ λ to each L-S path π of shape λ [11] for an appropriate ` (recall that an L-S path can be characterized by a collection of extremal weights and rational numbers). To construct a basis of H0(G/B,Lλ) = V ∗ λ , we use the contraction map to embed Vλ into (Mλ) ⊗`. Denote by pπ the image of bπ in V ∗ λ under the dual map (M ∗ λ) ⊗` → V ∗ λ . We show that the vectors pπ, π an L-S path of shape λ, form a basis of V ∗ λ . Further, the `-th power pπ ∈ H0(G/B,L`λ) is a product of extremal weight vectors pν1 · · · pν` , pνi ∈ H0(G/B,Lλ), plus a linear combination of elements which are “bigger” in some partial order. The basis given by the pπ is compatible with the restriction map H0(G/B,Lλ) → H0(X,Lλ) to a Schubert variety X , and it has the “standard monomial property”.

Tensor product structure of affine Demazure modules and limit constructions

Let g be a simple complex Lie algebra, we denote by ĝ the affine Kac-Moody algebra associated to the extended Dynkin diagram of g . Let Λ 0 be the fundamental weight of ĝ corresponding to the additional node of the extended Dynkin diagram. For a dominant integral g -coweight λ ∨ , the Demazure submodule V _ λ∨ (mΛ 0 ) is a g -module. We provide a description of the g -module structure as a tensor product of “smaller” Demazure modules. More precisely, for any partition of λ ∨ = λ∑ j as a sum of dominant integral g -coweights, the Demazure module is (as g -module) isomorphic to ⊗ j V _ (mΛ 0 ). For the “smallest” case, λ ∨ = ω ∨ a fundamental coweight, we provide for g of classical type a decomposition of V_ ω∨ (mΛ 0 ) into irreducible g-modules, so this can be viewed as a natural generalization of the decomposition formulas in [13] and [16]. A comparison with the U q (g)-characters of certain finite dimensional -modules (Kirillov-Reshetikhin-modules) suggests furthermore that all quantized Demazure modules V_ λ∨,q (mΛ 0 ) can be naturally endowed with the structure of a -module. We prove, in the classical case (and for a lot of non-classical cases), a conjecture by Kashiwara [10], that the “smallest” Demazure modules are, when viewed as g -modules, isomorphic to some KR-modules. For an integral dominant ĝ-weight Λ let V ( Λ ) be the corresponding irreducible ĝ-representation. Using the tensor product decomposition for Demazure modules, we give a description of the g -module structure of V ( Λ ) as a semi-infinite tensor product of finite dimensional g -modules. The case of twisted affine Kac-Moody algebras can be treated in the same way, some details are worked out in the last section.

Standard Monomial Theory and applications

In these notes, we explain how one can construct Standard Monomial Theory for reductive algebraic groups by using the path models of their representations and quantum groups at a root of unity. As applications, we obtain a combinatorial proof of the Demazure character formula and representation theoretic proofs of geometrical properties of Schubert varieties, such as normality, vanishing theorems, ideal theory and so on. Further applications of Standard Monomial Theory are made to prove geometrical properties of certain ladder determinantal varieties and certain quiver varieties. We sketch at the end an extension of the theory to Bott-Samelson varieties and configuration varieties.

Superbosonization of Invariant Random Matrix Ensembles

Abstract‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group Un, On, or USpn, giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s n-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.

Koreguläre und äquidimensionale Darstellungen

Zusammenfassung The aim of this note is to give a classification of all irreducible representations V of a connected semisimple algebraic group G such that either the quotient V ∥ G is smooth or the quotient map π: V → V ∥ G has equidimensional fibres. It follows by the classification that if the quotient map π is equidimensional, then the quotient V ∥ G is Smooth.

The Path Model for Representations of Symmetrizable Kac-Moody Algebras

In the theory of finite-dimensional representations of complex reductive algebraic groups, the group GL n (ℂ) is singled out by the fact that besides the usual language of weight lattices, roots, and characters, there exists an additional important combinatorial tool: the Young tableaux. For example, the sum over the weights of all tableaux of a fixed shape is the character of the corresponding representation, and the Littlewood-Richardson rule describes the decomposition of tensor products of GL n (ℂ)-modules purely in terms of the combinatoric of these Young tableaux. The advantage of this type of formula is (for example compared to Steinberg’s formula to decompose tensor products) that there is no cancellation of terms. This makes it much easier (and sometimes even possible) to prove for example that certain representations occur in a given tensor product.