Rational tableaux and the tensor algebra of gln

A generalization of the usual column-strict tableaux (equivalent to a construction of R. C. King) is presented as a natural combinatorial tool for the study of finite dimensional representations of GL_n(C). These objects are called rational tableaux since they play the same role for rational representations of GL_n as ordinary tableaux do for polynomial representations. A generalization of Schensted's insertion algorithm is given for rational tableaux, and is used to count the. multiplicities of the irreducible GL_n-modules in the tensor algebra of GL_n. The problem of counting multiplicities when the kth tensor power gl_n^⊗k is decomposed into modules which are simultaneously irreducible with respect to GL_n and the symmetric group S_k is also considered. The existence of an insertion algorithm which describes this decomposition is proved. A generalization of border strip tableaux, in which both addition and deletion of border strips is allowed, is used to describe the characters associated with this decomposition. For large n, these generalized border strip tableaux have a simple structure which allows derivation of identities due to Hanlon and Stanley involving the (large n) decomposition of gl_n^⊗k.     

A New Proof of the Mullineux Conjecture

Let S _d denote the symmetric group on d letters. In 1979 Mullineux conjectured a combinatorial algorithm for calculating the effect of tensoring with an irreducible S _d-module with the one dimensional sign module when the ground field has positive characteristic. Kleshchev proved the Mullineux conjecture in 1996. In the present article we provide a new proof of the Mullineux conjecture which is entirely independent of Kleshchev's approach. Applying the representation theory of the supergroup GL(m | n) and the supergroup analogue of Schur-Weyl Duality it becomes straightforward to calculate the combinatorial effect of tensoring with the sign representation and, hence, to verify Mullineux's conjecture. Similar techniques also allow us to classify the irreducible polynomial representations of GL(m | n) of degree d for arbitrary m, n, and d.     

Representations for eigenfunctions of expected value operators in the Wishart distribution

Recent articles by Kushner and Meisner (1980) and Kushner, Lebow and Meisner (1981) have posed the problem of characterising the ‘EP’ functions f(S) for which Ef(S) for which E(f(S)) = λ_nf(Σ) for some λ_n ? $\text{R}$, whenever the m × m matrix S has the Wishart distribution W(m, n, Σ). In this article we obtain integral representations for all nonnegative EP functions. It is also shown that any bounded EP function is harmonic, and that EP polynomials may be used to approximate the functions in certain L^p spaces.     

Theorem about the conjugacy representation ofS n

Every group has two natural representations on itself, the regular representation and the conjugacy representation. We know everything about the construction of the regular representation, but we know very little about the conjugacy representation (for uncommutative groups). In this paper we will see that every irreducible complex character ofS _n (n>2) is a constituent of conjugacy character ofS _n .     

Representation and character varieties of the Baumslag-Solitar groups

Representation and character varieties of the Baumslag–Solitar groups BS(p, q) are analyzed. Irreducible components of these varieties are found, and their dimension is calculated. It is proved that all irreducible components of the representation variety R_n(BS(p, q)) are rational varieties of dimension n², and each irreducible component of the character variety X_n(BS(p, q)) is a rational variety of dimension k ≤ n. The smoothness of irreducible components of the variety R_n^s (BS(p, q)) of irreducible representations is established, and it is proved that all irreducible components of the variety R_n^s (BS(p, q)) are isomorphic to A¹ {0}.     

An analysis of the multiplicity spaces in branching of symplectic groups

Branching of symplectic groups is not multiplicity free. We describe a new approach to resolving these multiplicities that is based on studying the associated branching algebra B{\mathcal{B}}. The algebra B{\mathcal{B}} is a graded algebra whose components encode the multiplicities of irreducible representations of Sp _2n–2 in irreducible representations of Sp _2n . Our first theorem states that the map taking an element of Sp _2n to its principal n × (n + 1) submatrix induces an isomorphism of B{\mathcal{B}} to a different branching algebra B^￠{\mathcal{B}^{\prime}}. The algebra B^￠{\mathcal{B}^{\prime}} encodes multiplicities of irreducible representations of GL _n–1 in certain irreducible representations of GL _n+1. Our second theorem is that each multiplicity space that arises in the restriction of an irreducible representation of Sp _2n to Sp _2n–2 is canonically an irreducible module for the n-fold product of SL ₂. In particular, this induces a canonical decomposition of the multiplicity spaces into one-dimensional spaces, thereby resolving the multiplicities.     

A Representation Stability Theorem for VI-modules

Let VI be the category whose objects are the finite dimensional vector spaces over a finite field of order q and whose morphisms are the injective linear maps. A VI-module over a ring is a functor from the category VI to the category of modules over the ring. A VI-module gives rise to a sequence of representations of the finite general linear groups. We prove that the sequence obtained from any finitely generated VI-module over an algebraically closed field of characteristic zero is representation stable - in particular, the multiplicities which appear in the irreducible decompositions eventually stabilize. We deduce as a consequence that the dimension of the representations in the sequence {V _n } obtained from a finitely generated VI-module V over a field of characteristic zero is eventually a polynomial in q ⁿ . Our results are analogs of corresponding results on representation stability and polynomial growth of dimension for FI-modules (which give rise to sequences of representations of the symmetric groups) proved by Church, Ellenberg, and Farb.     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

Branching laws for minimal holomorphic representations

In this paper we study the branching law for the restriction from SU(n,m) to SO(n,m) of the minimal representation in the analytic continuation of the scalar holomorphic discrete series. We identify the group decomposition with the spectral decomposition of the action of the Casimir operator on the subspace of S(O(n)×O(m))-invariants. The Plancherel measure of the decomposition defines an L²-space of functions, for which certain continuous dual Hahn polynomials furnish an orthonormal basis. It turns out that the measure has point masses precisely when n−m>2. Under these conditions we construct an irreducible representation of SO(n,m), identify it with a parabolically induced representation, and construct a unitary embedding into the representation space for the minimal representation of SU(n,m).     

Unitary Representations of Oligomorphic Groups

We obtain a complete classification of the continuous unitary representations of oligomorphic permutation groups (those include the infinite permutation group S ^??, the automorphism group of the countable dense linear order, the homeomorphism group of the Cantor space, etc.). Our main result is that all irreducible representations of such groups are obtained by induction from representations of finite quotients of open subgroups and, moreover, every representation is a sum of irreducibles. As an application, we prove that many oligomorphic groups have property (T). We also show that the Gelfand?CRaikov theorem holds for topological subgroups of S ^??: for all such groups, continuous irreducible representations separate points in the group.     

Irreducible representations of the full linear group

We obtain some results on Young diagrams. Based on these results, we construct bases for Young symmetry classes of tensors. Using these bases, we obtain a complete reduction of the representation A ??^mA [A∈GL(n,$\text{C}$] and irreducible matrix representations of the full linear group.     

Rational representations of Yangians associated with skew Young diagrams

Consider the general linear group GL_M over the complex field. The irreducible rational representations of the group GL_M can be labeled by the pairs of partitions and such that the total number of non-zero parts of and does not exceed M. Let EQ4 be the irreducible representation corresponding to such a pair. Regard the direct product as a subgroup of GL_N+M . Take any irreducible rational representation of GL_N+M. The vector space comes with a natural action of the group GL_N. Put n=. For any pair of standard Young tableaux of skew shapes respectively, we give a realization of as a subspace in the tensor product of n copies of defining representation of GL_N, and of ñ copies of the contragredient representation ()^*. This subspace is determined as the image of a certain linear operator on W_nñn. We introduce this operator by an explicit multiplicative formula. When M=0 and is an irreducible representation of GL_N, we recover the known realization of as a certain subspace in the space of all traceless tensors in . Then the operator may be regarded as the rational analogue of the Young symmetrizer, corresponding to the tableau of shape . Even when M=0, our formula for is new. Our results are applications of the representation theory of the Yangian of the Lie algebra . In particular, is an intertwining operator between certain representations of the algebra on . We also introduce the notion of a rational representation of the Yangian . As a representation of , the image of is rational and irreducible.Mathematics Subject Classification (2000): 17B37, 20C30, 22E46, 81R50in final form: 10 July 2003     

REPRESENTATION THEORY IN COMPLEX RANK,I

P. Deligne defined interpolations of the tensor category of representations of the symmetric group S _n to complex values of n. Namely, he defined tensor categories Rep(S _t ) for any complex t. This construction was generalized by F. Knop to the case of wreath products of S _n with a finite group. Generalizing these results, we propose a method of interpolating representation categories of various algebras containing S _n (such as degenerate affine Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, etc.) to complex values of n. We also define the group algebra of S _n for complex n, study its properties, and propose a Schur-Weyl duality for Rep(S _t ).     

Semidefinite descriptions of low-dimensional separable matrix cones

Let K⊂E, K^′⊂E^′ be convex cones residing in finite-dimensional real vector spaces. An element y in the tensor product E⊗E^′ is K⊗K^′-separable if it can be represented as finite sum , where x_l∈K and for all l. Let S(n), H(n), Q(n) be the spaces of n×n real symmetric, complex Hermitian and quaternionic Hermitian matrices, respectively. Let further S₊(n), H₊(n), Q₊(n) be the cones of positive semidefinite matrices in these spaces. If a matrix A∈H(mn)=H(m)⊗H(n) is H₊(m)⊗H₊(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m)⊗S(n), H(m)⊗S(n), and for m?2 in the space Q(m)⊗S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n)⊗S(2) is Q₊(n)⊗S₊(2)- separable if and only if it is positive semidefinite.     

对称群S5的表示环

本文利用有限群特征标理论计算了对称群S₅的所有不可约复表示的幂公式.根据求解幂公式过程中得到的S₅任意两个不可约表示张量积的分解情况,作者刻画了S₅上表示环r(S₅)及其若干结构性质,如极小生成元关系式表达、单位群、本原幂等元、行列式与Casimir数.     

Double affine Hecke algebras,conformal coinvariants and Kostka polynomials

We study a class of representations called ‘calibrated representations’ of the rational and trigonometric double affine Hecke algebras of type ${\mathit{GL}}_n$. We give a realization of calibrated irreducible modules as spaces of coinvariants constructed from integrable modules over the affine Lie algebra ${\stackrel{?}{\mathfrak{gl}}}_m$. We also give a character formula of these irreducible modules in terms of a generalization of Kostka polynomials. These results are conjectured by Arakawa, Suzuki and Tsuchiya based on the conformal field theory. The proofs using recent results on the representation theory of the double affine Hecke algebras will be presented in the forthcoming papers. To cite this article: T. Suzuki, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Criteria for the compactness and the Fredholm property of pseudodifferential operators in Hölder-Zygmund weighted spaces

We obtain necessary and sufficient conditions for the complete continuity (the Fredholm property) in Hölder-Zygmund spaces on ? ⁿ whose weight has a power-law behavior at infinity for pseudodifferential operators with symbols in the Hörmander class S _1,δ ^m , 0 ≤ δ < 1 (slowly varying symbols in the class S _1,0 ^m ). We show that such operators are compact operators or Fredholm operators in weighted Hölder-Zygmund spaces if and only if they are compact operators or Fredholm operators, respectively, in Sobolev spaces.