Related papers: A virtual character and nonzero Kronecker coeffici…

Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest well-suited counterparts of the Kerov polynomials in spin (or projective)…

Let k be a non archimedean field. If X is a k-algebraic variety and U a locally closed semi-algebraic subset of X^{an} -- the Berkovich space associated to X -- we show that for l \neq char(\tilde{k}), the cohomology groups H^i_c (\bar{U},…

In 2015, the author proved combinatorially character formulas expressing sums of the (formal) dimensions of irreducible representations of symplectic groups, refining some works of Nekrasov and Okounkov, Han, King, and Westbury. In this…

Let $G$ be a finite group and let $\rm{Irr}(G)$ be the set of all irreducible complex characters of $G$. For a character $\chi \in \rm{Irr}(G)$, the number $\rm{cod}(\chi):=|G:\rm{ker}\chi|/\chi(1)$ is called the co-degree of $\chi$. The…

Let $S_n$ denote a symmetric group, $\chi$ an irreducible character of $S_n$, and $g\in S_n$ an element which decomposes into $k$ disjoint cycles, where $1$-cycles are included. Then $|\chi(g)|\le k!$, and this upper bound is sharp for…

The Littlewood-Richardson coefficients $c^\nu_{\lambda,\mu}$ are the multiplicities in the tensor product decomposition of two irreducible representations of the general linear group GL$(n, {\mathbb C})$. They are parametrized by the…

Let K $\subset$ G be compact connected Lie groups with common maximal torus T. Let (M, $\omega$) be a prequantisable compact connected symplectic manifold with a Hamiltonian G-action. Geometric quantisation gives a virtual representation of…

For any positive integers $s$ and $t$, let $Q_{t}^{s}(n)$ denotes the number of partitions of a positive integer $n$ into distinct parts such that no part is congruent to $s$ or $t-s$ modulo $t$. We prove some Ramanujan-type congruences for…

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of…

Given an integer $n\geq 1$ and an irreducible character $\chi_{\lambda}$ of $S_{n}$ for some partition $\lambda$ of $n$, the immanant $\mathrm{imm}_{\lambda}:\mathbb{C}^{n\times n}\to\mathbb{C}$ maps matrices $A\in\mathbb{C}^{n\times n}$ to…

We consider problems concerning the largest degrees of irreducible characters of symmetric groups, and the multiplicities of character degrees of symmetric groups. Using evidence from computer experiments, we posit several new conjectures…

We present new proofs and generalizations of unimodality of the q-binomial coefficients \binom{n}{k}_q as polynomials in q. We use an algebraic approach by interpreting the differences between numbers of certain partitions as Kronecker…

Motivated by Stanley's results in \cite{St02}, we generalize the rank of a partition $\lambda$ to the rank of a shifted partition $S(\lambda)$. We show that the number of bars required in a minimal bar tableau of $S(\lambda)$ is max$(o, e +…

For the BGG category of $\mathfrak{q}(n)$-modules of half-integer weights, a Kazhdan-Lusztig conjecture \`a la Brundan is formulated in terms of categorical canonical basis of the $n$th tensor power of the natural representation of the…

An elegant but elementary result of Wolff from 1921, when interpreted in terms of Banach algebras, shows that it is possible to find a sequence of distinct characters $\phi_n$ on the disc algebra and an $\ell_1$ sequence of complex numbers…

The problem of computing the characters of the finite dimensional irreducible representations of the Lie superalgebra $\mathfrak q(n)$ over $\C$ was solved in 1996 by I. Penkov and V. Serganova. In this article, we give a different approach…

In 2013, Pak and Panova proved the strict unimodality property of $q$-binomial coefficients $\binom{\ell+m}{m}_q$ (as polynomials in $q$) based on the combinatorics of Young tableaux and the semigroup property of Kronecker coefficients.…

Cauchy summation formula plays a central role in application of character calculus to many problems, from AGT-implied Nekrasov decomposition of conformal blocks to topological-vertex decompositions of link invariants. We briefly review the…

It is well known that the number of real irreducible characters of a finite group G coincides with the number of real conjugacy classes of G. Richard Brauer has asked if the number of irreducible characters with Frobenius-Schur indicator 1…

In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with…