Related papers: Plethysm and a character embedding problem of Mill…

The object of this short note is to prove a theorem and present a conjecture for the number of even entries in the character table of the symmetric group.

We compute dimensions of the components for the operad of two compatible brackets and for the bihamiltonian operad. We also obtain character formulas for the representations of the symmetric groups and the $SL_2$ group in these spaces.

Plethysm of two Schur functions can be expressed as a linear combination of Schur functions, and monomial symmetric functions. In this paper, we express the coefficients combinatorially in the case of monomial symmetric functions. And by…

In this paper, we prove Lusztig's conjecture for finite special linear groups, i.e., we show that characteristic functions of character sheaves coincide with almost characters up to scalar constants, under the condition that the…

The classification of irreducible, spherical characters of the infinite-dimensional unitary/orthogonal/symplectic groups can be obtained by finding all possible limits of normalized, irreducible characters of the corresponding…

A new type of conjectures on characters of finite groups, related to the McKay conjecture, have recently been proposed. In this paper, we study these conjectures for symmetric groups.

For any set representation (permutation representation) of the symmetric group $S_n$, we give combinatorial interpretation for coefficients of its Frobenius character expanded in the basis of monomial symmetric functions.

We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.

We prove an identity for Littlewood--Richardson coefficients conjectured by Pelletier and Ressayre (arXiv:2005.09877). The proof relies on a novel birational involution defined over any semifield.

We derive group branching laws for formal characters of subgroups $H_\pi$ of GL(n) leaving invariant an arbitrary tensor $T^\pi$ of Young symmetry type $\pi$ where $\pi$ is an integer partition. The branchings $GL(n)\downarrow GL(n-1)$,…

This paper proves a combinatorial rule expressing the product $s_\tau(s_{\lambda/\mu} \circ p_r)$ of a Schur function and the plethysm of a skew Schur function with a power sum symmetric function as an integral linear combination of Schur…

We give a definition of character sheaves on the group compactification which is equivalent to Lusztig's definition in \cite{L3}. We also prove some properties of the character sheaves on the group compactification.

In this paper we study the Frobenius characters of the invariant subspaces of the tensor powers of a representation V. The main result is a formula for these characters for a polynomial functor of V involving the characters for V. The main…

Computations of Miller and Scheinerman suggest that the vast majority of the zeros appearing in the character table of the symmetric group are of a certain special type. While we cannot prove this, we resolve a conjecture arising in their…

We prove a recursive formula for plethysm coefficients of the form $a^\mu_{\lambda,(m)}$, generalising results on plethysms due to Bruns--Conca--Varbaro and de Boeck--Paget--Wildon. From this we deduce a stability result and resolve two…

Stanley introduces polynomials which help evaluate symmetric group characters and conjectures that the coefficients of the polynomials are positive. Stanley later gives a conjectured combinatorial interpretation for the coefficients of the…

We obtain transformation formulas for quadratic character sums with quartic and cubic polynomial arguments.

The plethysm product of Schur functions corresponds to composing polynomial representations of infinite general linear groups. Finding the plethysm coefficients $\langle s_\nu \circ s_\mu, s_\lambda\rangle$ that express an arbitrary…

We give a simple proof of a recent result by J. Schleischitz dealing with a counterexample to the uniform Littlewood conjecture. Our construction is based on simple properties of Fibonacci numbers.

We generalize a result by Cunningham-Salmasian to a Mackey-type formula for the compact restriction of a semisimple perverse sheaf produced by parabolic induction from a character sheaf, under certain conditions on the parahoric group used…