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We prove that an N-state vertex model representation of the An Hecke algebra quotient NHn(q) is faithful for all q. We use the result to examine the indecomposable content of these representations, and hence the structure of the centraliser algebra, which is generically a quotient of Uqsl(N), at q a root of unity. We achieve a complete analysis in the case N=2, finding a number of Morita self-dual algebraic structures.

We present an inductive method for constructing the basic spin representations of the double covers of the symmetric groups over fields of any characteristic., 8 pages

In this chapter, we discuss the representation theory of GL n (ℂ). Schur modules are introduced and are shown to be irreducible GL n (ℂ)-modules, using Schur-Weyl duality. We then discuss the representation theory of SL n (ℂ), and deduce the representation theory of GL n (ℂ). For further details, refer to [17].

In this paper we investigate a multi-parameter deformation $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ of the walled Brauer algebra which was previously introduced by Leduc (\cite{leduc}). We construct an integral basis of $\mathfrak{B}_{r,s}^n(a,\lambda,\delta)$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $\mathfrak{B}_{r,s}^n(q)= \mathfrak{B}_{r,s}^n(q^{-1}-q,q^n,[n]_q)$ on mixed tensor space and prove that the kernel is free over the ground ring $R$ of rank independent of $R$. As an application, we prove one side of Schur--Weyl duality for mixed tensor space: the image of $\mathfrak{B}_{r,s}^n(q)$ in the $R$-endomorphism ring of mixed tensor space is, for all choices of $R$ and the parameter $q$, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra $\mathbf{U}$ of the general linear Lie algebra $\mathfrak{gl}_n$ on mixed tensor space. Thus, the $\mathbf{U}$-invariants in the ring of $R$-linear endomorphisms of mixed tensor space are generated by the action of $\mathfrak{B}_{r,s}^n(q)$., Comment: 34 pages

A class of mathematical dualities have played a central role in mapping states in gauge theory to states in the spacetime string theory dual. This includes the classical Schur-Weyl duality between symmetric groups and Unitary groups, as well as generalisations involving Brauer and Hecke algebras. The physical string dualities involved include examples from the AdS/CFT correspondence as well as the string dual of two-dimensional Yang Mills., 10 pages,talk given at ``Ten years of AdS/CFT,'' Buenos Aires, December 19-21, 2007

For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs., 9 pages, 6 figures To appear in DMTCS as part of the FPSAC 2008 conference proceedings

We establish a convergent power series expansion for the expectation of a product of traces of powers of a random unitary matrix under the heat kernel measure. These expectations turn out to be the generating series of certain paths in the Cayley graph of the symmetric group. We then compute the asymptotic distribution of a random unitary matrix under the heat kernel measure on the unitary group U(N) as N tends to infinity, and prove a result of asymptotic freeness result for independent large unitary matrices, thus recovering results obtained previously by Xu and Biane. We give an interpretation of our main expansion in terms of random ramified coverings of a disk. Our approach is based on the Schur-Weyl duality and we extend some of our results to the orthogonal and symplectic cases.

The Schur-Weyl duality provides a systematic approach to the study of representations of the classical groups. In this contribution we use the Schur-Weyl duality to analyze the natural representations of the symmetric group S n arising from the irreducible GL(n, C) and SL(n, C)-modules. This paper is dedicated to A. A. Kirillov on the occasion of his 26-th birthday.

The notion of two-sided cell, which was originally introduced by A. Joseph and reformulated by D. Kazhdan and G. Lusztig, has played an important role in the representation theory. Results concerning them have been obtained by very deep and sometimes ad hoc arguments. Here we introduce certain polynomial invariants for irreducible representations of Weyl groups. Our invariants are easily calculated, and the calculational results show that they almost determine the two-sided cells. Moreover, the factorization pattern of our polynomial invariants seems to be controlled by the natural parameter set $\mathscr{M}(\mathscr{G})$ of each two-sided cell.

Publisher Summary This chapter highlights the "classical" representation theory for reductive algebraic groups over algebraically closed fields. The theory is developed in such a way that almost everything carries over to quantum groups. It emphasizes the similarities between the modular representation theory and the representation theory for quantum groups at roots of unity. The chapter provides basic general definitions concerning algebraic groups and their representations. In addition, it also proves the Borel-Weil-Bott theorem and Serre duality. From these two results it is easily deduced the complete reducibility of all finite dimensional modules for algebraic groups in characteristic zero. Furthermore, it illustrates a sketch how the representation theory of quantum groups behaves very similarly to the representation theory for reductive algebraic groups.

Duality and the use of the geometry caused by duality are powerful tools in representation theory. A very nice example for this is Okuyama’s proof that G has a normal Sylow 2-subgroup if 2 does not divide the dimension of any absolutely simple module in characteristic 2 (see [12]).

The concept of quantum groups is important for the study of the quantum Yang-Baxter equations, Drinfeld [2], Jimbo [4], Manin [5] and others. On the other hand, Woronowicz [10] introduced the concept of compact matrix pseudogroups through the study of the dual object of groups. As pointed out by Rosso in [8], these two concepts are related to each other as quantum Lie algebras and quantum Lie groups. In this talk we want to indicate that the ideas of Kac algebras studied by Takesaki [9] and Enock and Schwartz [3] et al. are helpful for the study of quantum groups. As a result we can give a geometric interpretation for a q-analogue of a certain class of special functions, which has been a long standing problem of q- analogues.

An extension of the Schur-Weyl duality connecting the representations of the symmetric and unitary groups is given. The Schur-Weyl basis is constructed using annihilation and creation operators. Three factorisation lemmas are derived. Their importance lies in the fact that they relate the phase freedoms within the Racah-Wigner algebra of the symmetric groups and the unitary groups. Extensions of the Regge symmetries are also given. These are expressed in five duality relations.

A unitary tensor product representation of the group S O0(1,n), n =2,3,4, does not contain the trivial representation as a discrete direct summand unless each of the factors does.

We have long suspected the existence of two theorems about the decomposition matrices of the symmetric groups which involve the removal of the first row or the first column from a diagram. The purpose of this paper is to state and prove these theorems. It is interesting to note that we have been unable to construct proofs which lie entirely within the framework of the representation theory of symmetric groups, and the symmetric groups results are here deduced from two stronger theorems concerning the representation theory of the general linear group.

We describe an efficient method for the computer evaluation of the ordinary irreducible polynomial representations of general linear groups using an integral form of the ordinary irreducible representations of symmetric groups. In order to do this, we first give an algebraic explanation of D. E. Littlewood's modification of I. Schur's construction. Then we derive a formula for the entries of the representing matrices which is much more concise and adapted to the effective use of computer calculations. Finally, we describe how one obtains — using this time an orthogonal form of the ordinary irreducible representations of symmetric groups — a version which yields a unitary representation when it is restricted to the unitary subgroup. In this way we adapt D. B. Hunter's results which heavily rely on Littlewood's methods, and boson polynomials come into the play so that we also meet the needs of applications to physics.

Those unitary representations of SL4$(R)bar)= are classified which contain each irreducible SŌ(4)bar)= representation at most once.

Publisher Summary This chapter presents an overview of a characteristic-free approach to the representation theory of the general linear and symmetric groups along letter place algebras. Mackey's intertwining number theorem suggests to watch for a second R -basis of R αβ , which is of representation theoretical relevance. According to Young's rule, such a basis should be closely connected with the standard bitableaux of content ( α, β ). The chapter presents a characteristic-free construction of the irreducible modules for the general linear and symmetric groups in terms of symmetrized bideterminants. This construction is handier than the known descriptions of these modules.

Let r > 0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauer's and Robinson's solution of Nakayama's conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r > 2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible characters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height h t z = 0 . Therefore Alperin's conjecture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspondences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-subgroup /~-~R such that there is a natural height preserving one-to-one correspondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b).

$1. Totally Geodesic Submanifolds of Xand of S. Reduction to the Hermitian Hyperbolic Case $2. Conical Distributions. Elementary Properties $3. Conical Functions and Finite-Dimensional Representations ...... $4. The Determination of the Conical Distributions $5. The Spherical Representations and the Fourier Transform on X .... 96. Construction of the Intertwining Operators

We obtain Schur–Weyl dualities in which the algebras, acting on both sides, are semigroup algebras of various symmetric inverse semigroups and their deformations.

We give a closed formula for the trace of the partition algebra P"n(x) acting on an irreducible representation whose basis is indexed by the set partitions of {1,...,n}. This trace counts the set partitions that are fixed under the action of the symmetric group S"n. We use this trace to determine an analog of the ''second orthogonality of characters'' formula for P"n(x) and to compute the trace of the biregular representation of P"n(x). We use the Schur-Weyl duality between P"n(r) and the symmetric group S"r to study fixed points of random permutations in S"r. In particular, we compute the joint mixed moments of the random variables Tr(@s^j) for @s@?S"r.

We prove gap results for the low-dimensional representation of unitary groups in nondefining characteristics. More precisely, we give a lower bound for the third smallest degree of a nontrivial absolutely irreducible representation of the groups mentioned above, which is almost in the order of magnitude of the square of the two smallest such representations. As a corollary we obtain the uniqueness of Weil representations in all cross characteristics for the unitary groups. Our approach uses some results on ordinary representations which can be proved using Deligne–Lusztig theory and which might be of independent interest, as well as information on decomposition matrices.

We compute a stable formula for the Hilbert series of the invariant algebra of polynomial functions on @?"i"="1^rC^n^"^i under the action of U(n"1)x...xU(n"r), when viewed as real vector space. This situation has a physical interpretation as it is the quantum analog of an r-particle classical system in which the ith particle has n"i classical states. The stable formula involves only elementary combinatorics, while its derivation involves the representation theory of the symmetric group. In particular, the Kronecker coefficients play an important role.

The purpose of this paper is to look at some results in the representation theory of the symmetric groups, both old and recent, from a modern point of view. In the first two sections we construct the irreducible representations of the symmetric groups as left ideals in the group ring. The irreducible modular representations were first constructed by James [ 131. We prove inequivalence by a method more natural than the original. The approach may be compared with that in [6]. A comprehensive survey of recent developments has been published by their originator [ 151. In Section 3 we look at the skew representations. Their dimension may be established by the same method as that used to obtain the dimenssions of the Specht modules in [lo]. A further study of skew representations may be found in [ 161. The induction algebra of symmetric groups is defined in Section 4. It is proved to be a polynomial algebra, a result also obtained in [9]. The algebra is used to give a new proof of Chung’s conjecture that the number of modular irreducible representations in a block is independent of the core. In the last two sections we examine Taulbee’s construction [7] of a class of projective, indecomposable characters. The construction uses the process of r-induction (which has been used to calculate decomposition numbers of

The Adler–Bell–Jackiw (ABJ) anomalies for all representations of the classical groups are calculated with the aid of a technique proposed recently for the representation theory of these groups. The method presented here is useful also for the calculations of the eigenvalues of Casimir invariants with rank higher than 3.

Polynomial Representations of GLn(K): The Schur algebra.- Weights and Characters.- The modules D?,K.- The Carter-Lusztig modules V?,K.- Representation theory of the symmetric group.

1. Symmetric groups and their young subgroups 2. Ordinary irreducible representations and characters of symmetric and alternating groups 3. Ordinary irreducible matrix representations of symmetric groups 4. Representations of wreath products 5. Applications to combinatories and representation theory 6. Modular representations 7. Representation theory of Sn over an arbitrary field 8. Representations of general linear groups Appendices Index.