Related papers: Symmetric group characters as symmetric functions
We consider a filtration of the symmetric function space given by $\Lambda^{(k)}_t$, the linear span of Hall-Littlewood polynomials indexed by partitions whose first part is not larger than $k$. We introduce symmetric functions called the…
We study the restriction to Sylow subgroups of irreducible characters of symmetric groups. In particular, we focus our attention on constituents of degree greater than 1. Our main result is a wide generalization of Theorem 3.1 of Giannelli…
Basis functions which are invariant under the operations of a rotational point group $G$ are able to describe any 3-D object which exhibits the rotational point group symmetry. However, in order to characterize the spatial statistics of an…
The decomposition matrix of a finite group in prime characteristic p records the multiplicities of its p-modular irreducible representations as composition factors of the reductions modulo p of its irreducible representations in…
Let K be an algebraically closed field of characteristic p>0 and let Sp(2m) be the symplectic group of rank m over K. The main theorem of this article gives the character of the rational simple Sp(2m)-modules with fundamental highest weight…
We give another proof, using tools from Geometric Invariant Theory, of a result due to S. Sam and A. Snowden in 2014, concerning the stability of Kro-necker coefficients. This result states that some sequences of Kronecker coefficients…
The paper investigates the stability properties of restrictions of irreducible representations of the symmetric group to the hyperoctahedral subgroup. A stability result is obtained, analogous to the classical Murnaghan theorem on the…
In this paper we construct an algebraic invariant attached to Galois representations over number fields. This invariant, which we call an Artin symmetric function, lives in a certain ring we introduce called the ring of arithmetic symmetric…
We define an abelian group homomorphism $\mathscr{F}$, which we call the Frobenius transform, from the ring of symmetric functions to the ring of the symmetric power series. The matrix entries of $\mathscr{F}$ in the Schur basis are the…
Representation theory of the symmetric group $\mathfrak{S}_n$ has a very distinctive combinatorial flavor. The conjugacy classes as well as the irreducible characters are indexed by integer partitions $\lambda \vdash n$. We introduce class…
In previous work Regev used part of the representation theory of Lie superalgebras to compute the values of a character of the symmetric group whose decomposition into irreducible constituents is described by semistandard…
We consider an operator of Bernstein for symmetric functions, and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the…
The aim of this paper is to describe structural properties of spaces of diagonal rectangular harmonic polynomials in several sets (say $k$) of $n$ variables, both as $GL_k$-modules and $S_n$-modules. We construct explicit such modules…
Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with $K$-theory yields a rich combinatorial theory of inhomogeneous deformations, where…
A known result for the finite general linear group $\GL(n,\FF_q)$ and for the finite unitary group $\U(n,\FF_{q^2})$ posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group. Fulman…
We consider sequences of degrees of ordinary irreducible $S_n$-characters. We assume that the corresponding Young diagrams have rows and columns bounded by some linear function of $n$ with leading coefficient less than one. We show that any…
We obtain the Schur positivity of spider graphs of the forms $S(a,2,1)$ and $S(a,4,1)$, which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find…
How to study a nice function on the real line? The physically motivated Fourier theory technique of harmonic analysis is to expand the function in the basis of exponentials and study the meaningful terms in the expansion. Now, suppose the…
We prove a closed character formula for the symmetric powers $S^N V(\lambda)$ of a fixed irreducible representation $V(\lambda)$ of a complex semi-simple Lie algebra $\mathfrak{g}$ by means of partial fraction decomposition. The formula…
Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by…