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In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on…

We present a unified framework of combinatorial descriptions, and the analogous asymptotic growth of the coefficients of two general families of functions related to integer partitions. In particular, we resolve several conjectures and…

We prove a universal identity for powers of elements in quadratic algebras, expressing x^m in terms of x and the identity. As a consequence, we obtain a general formula for powers of 2x2 matrices depending only on trace and determinant.…

A general setting to study a certain type of formulas, expressing characters of the symmetric group $\mathfrak{S}_n$ explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is…

By exploiting relationships between the values taken by ordinary characters of symmetric groups we prove two theorems in the modular representation theory of the symmetric group. 1. The decomposition matrices of symmetric groups in odd…

We generalize geometric prequantization of symplectic manifolds to differentiable stacks. Our approach is atlas-independent and provides a bijection between isomorphism classes of principal circle bundles (with or without connections) and…

Rectangulations are partitions of a square into axis-aligned rectangles. A number of results provide bijections between combinatorial equivalence classes of rectangulations and families of pattern-avoiding permutations. Other results deal…

The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the…

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending…

The character table of the symmetric group $S_n$, of permutations of $n$ objects, is of fundamental interest in theoretical physics, combinatorics as well as computational complexity theory. We investigate the implications of an identity,…

We study a family of integrable systems of nonlinearly coupled harmonic oscillators on the classical and quantum levels. We show that the integrability of these systems follows from their symmetry characterized by algebras called here…

Universality of eigenvalue spacings is one of the basic characteristics of random matrices. We give the precise meaning of universality and discuss the standard universality classes (sine, Airy, Bessel) and their appearance in unitary,…

We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial…

We provide a new branching rule from the general linear group $GL_{2n}(\mathbb{C})$ to the symplectic group $Sp_{2n}(\mathbb{C})$ by establishing a simple algorithm which gives rise to a bijection from the set of semistandard tableaux of a…

Permutation Matrices are a well known class of matrices which encode the elements of the symmetric group on $d$ elements as a square $d\times d$ matrix. Motivated by [4], we define a similar class of matrices which are a generalization of…

Just as the definition of factorial Schur functions as a ratio of determinants allows one to show that they satisfy a Jacobi-Trudi-type identity and have an explicit combinatorial realisation in terms of semistandard tableaux, so we offer…

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kir\'aly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear…

In the work of Varchenko, Zagier, Thibon, and Reiner, Saliola, Welker, linear algebraic properties of the multiplication map on the group algebra of the group algebra element are studied, which is the sum over all permutations weighted by…

It is known that characters of irreducible representations of finite Lie algebras can be obtained using theWeyl character formula including Weyl group summations which make actual calculations almost impossible except for a few Lie algebras…

We consider irreducible representations of finite quandles over $\mathbb{C}$. For $Q$ a finite quandle whose inner automorphism group $Inn(Q)$ have trivial Schur multipliers, we prove that the irreducible representations of $Q$ can be…