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We show that there exists a family of groups $G_n$ and nontrivial irreducible representations $rho_n$ such that, for any constant $t$, the average of $rho_n$ over $t$ uniformly random elements $g_1, ldots, g_t in G_n$ has operator norm $1$ with probability approaching 1 as $n rightarrow infty$. More quantitatively, we show that there exist families of finite groups for which $Omega(log log |G|)$ random elements are required to bound the norm of a typical representation below $1$. This settles a conjecture of A. Wigderson.
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We consider the asymptotic behavior as $ntoinfty$ of the spectra of random matrices of the form [frac{1}{sqrt{n-1}}sum_{k=1}^{n-1}Z_{nk}rho_n ((k,k+1)),] where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $rho_n$ of the symmetric group on ${1,2,...,n}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on ${1,2,...,n}$ are indexed by partitions $lambda_n$ of $n$. A consequence of the results we establish is that if $lambda_{n,1}gelambda_{n,2}ge...ge0$ is the partition of $n$ corresponding to $rho_n$, $mu_{n,1}gemu_{n,2}ge >...ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $mu_n$ is the transpose of the Young diagram of $lambda_n$), $lim_{ntoinfty}frac{lambda_{n,i}}{n}=p_i$ for each $ige1$, and $lim_{ntoinfty}frac{mu_{n,j}}{n}=q_j$ for each $jge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $theta Z$ and variance $1-theta^2$, where $theta$ is the constant $sum_ip_i^2-sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.
We give a description of the centralizer algebras for tensor powers of spin objects in the pre-modular categories $SO(N)_2$ (for $N$ odd) and $O(N)_2$ (for $N$ even) in terms of quantum $(n-1)$-tori, via non-standard deformations of $Umathfrak{so}_N$. As a consequence we show that the corresponding braid group representations are Gaussian representations, the images of which are finite groups. This verifies special cases of a conjecture that braid group representations coming from weakly integral braided fusion categories have finite image.
For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3. We then form the distributive lattices of order ideals of these posets. Corollary 5.4 gives elegant quotient-of-products expressions for the rank generating functions of these lattices (thereby providing answers to a 1979 question of Stanley). Also, Theorem 5.3 describes how these lattices provide a new combinatorial setting for the Weyl characters of representations of rank two semisimple Lie algebras. Most of these lattices are new; the rest of them (or related structures) have arisen in work of Stanley, Kashiwara, Nakashima, Littelmann, and Molev. In a future paper, one author shows that the posets constructed here form a Dynkin diagram-indexed answer to a combinatorially posed classification question. In a companion paper, some of these lattices are used to explicitly construct some representations of rank two semisimple Lie algebras. This implies that these lattices are strongly Sperner.
This document is a companion for the Maple program : Discrete series and K-types for U(p,q) available on:http://www.math.jussieu.fr/~vergne We explain an algorithm to compute the multiplicities of an irreducible representation of U(p)x U(q) in a discrete series of U(p,q). It is based on Blattners formula. We recall the general mathematical background to compute Kostant partition functions via multidimensional residues, and we outline our algorithm. We also point out some properties of the piecewise polynomial functions describing multiplicities based on Paradans results.
In this paper, we define a set which has a finite group action and is generated by a finite color set, a set which has a finite group action, and a subset of the set of non negative integers. we state its properties to apply one of solution of the following two problems, respectively. First, we calculate the generating function of the character of symmetric powers of permutation representation associated with a set which has a finite group action. Second, we calculate the number of primitive colorings on some objects of polyhedrons. It is a generalization of the calculation of the number of primitive necklaces by N.Metropolis and G-C.Rota.
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