The character and the wave front set correspondence in the stable range

We relate the distribution characters and the wave front sets of unitary representation for real reductive dual pairs of type I in the stable range.     

Generalized characters of the symmetric group

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair (S(n)×S(n),diagS(n)). They form an orthogonal basis in the space of the functions on the group S(n) invariant with respect to conjugations by S(n). In this paper we consider a different Gelfand pair connected with the symmetric group, that is an “unbalanced” Gelfand pair (S(n)×S(n−1),diagS(n−1)). Zonal spherical functions of this Gelfand pair form an orthogonal basis in a larger space of functions on S(n), namely in the space of functions invariant with respect to conjugations by S(n−1). We refer to these zonal spherical functions as normalized generalized characters of S(n). The main discovery of the present paper is that these generalized characters can be computed on the same level as the irreducible characters of the symmetric group. The paper gives a Murnaghan-Nakayama type rule, a Frobenius type formula, and an analogue of the determinantal formula for the generalized characters of S(n).     

On the Kernel of Restriction of Characters*

Let G be a finite group, H be a proper subgroup of G, and S be a unitary subring of C. The kernel of the restriction map S[Irr(G)] → S[Irr(H)] as a ring homomorphism is studied. As a corollary, the main result in [Isaacs, I. M. and Navarro, G., Injective restriction of characters, Arch. Math., 108, 2017, 437–439] is reproved.     

Quadratic characters in groups of odd order

We prove that in a finite group of odd order, the number of irreducible quadratic characters is the number of quadratic conjugacy classes.     

直接键连原子间的NMR偶合常数的自然杂化轨道研究V.~(13)C-~(13)C和~(13)C-~(31)P核自旋偶合常数~1J_(CC)和~1J_(CP)

在前文工作的基础上，结合ＭＮＤＯ／ＥＨＭＯ分子轨道方法和自然杂化轨道方法，具体计算了ＣＣ键和ＣＰ键的核自旋偶合常数．计算结果表明，１ＪＣＣ和１ＪＣＰ主要由成键原子的轨道杂化作用和键极性这两种结构因素所决定．为从简单价键理论角度解释和计算１ＪＣＣ和１ＪＣＰ值提供了简便直观的方法．     

Upper bound on the characters of the symmetric groups for balanced Young diagrams and a generalized Frobenius formula

We study asymptotics of an irreducible representation of the symmetric group Sn corresponding to a balanced Young diagram λ (a Young diagram with at most rows and columns for some fixed constant C) in the limit as n tends to infinity. We show that there exists a constant D (which depends only on C) with a property that

     

Characters and a Verlinde-type formula for symmetric Hopf algebras

We study certain aspects of finite-dimensional non-semisimple symmetric Hopf algebras H and their duals H^*. We focus on the set I(H) of characters of projective H-modules which is an ideal of the algebra of cocommutative elements of H^*. This ideal corresponds via a symmetrizing form to the projective center (Higman ideal) of H which turns out to be ΛH, where Λ is an integral of H and is the left adjoint action of H on itself. We describe ΛH via primitive and central primitive idempotents of H. We also show that it is stable under the quantum Fourier transform. Our best results are obtained when H is a factorizable ribbon Hopf algebra over an algebraically closed field of characteristic 0. In this case ΛH is also the image of I(H) under a “translated” Drinfel'd map. We use this fact to prove the existence of a Steinberg-like character. The above ingredients are used to prove a Verlinde-type formula for ΛH.     

Tightening Turyn’s bound for Hadamard difference sets

This work examines the existence of (4q ²,2q ²−q,q ²−q) difference sets, for q=p ^f , where p is a prime and f is a positive integer. Suppose that G is a group of order 4q ² which has a normal subgroup K of order q such that G/K ≅ C _q ×C ₂×C ₂, where C _q ,C ₂ are the cyclic groups of order q and 2 respectively. Under the assumption that p is greater than or equal to 5, this work shows that G does not admit (4q ²,2q ²−q,q ²−q) difference sets.     

More constructions of pseudorandom sequences of k symbols

We present some new constructions of families of pseudorandom sequences of k symbols, which generalize several previous constructions for the binary case.     

Stabilized plethysms for the classical Lie groups

The plethysms of the Weyl characters associated to a classical Lie group by the symmetric functions stabilize in large rank. In the case of a power sum plethysm, we prove that the coefficients of the decomposition of this stabilized form on the basis of Weyl characters are branching coefficients which can be determined by a simple algorithm. This generalizes in particular some classical results by Littlewood on the power sum plethysms of Schur functions. We also establish explicit formulas for the outer multiplicities appearing in the decomposition of the tensor square of any irreducible finite-dimensional module into its symmetric and antisymmetric parts. These multiplicities can notably be expressed in terms of the Littlewood-Richardson coefficients.