Related papers: Symmetric group characters as symmetric functions …

Denote the symmetric group of degree $n$ by $S_n$. Let $\rho$ be an irreducible representation of $S_n$ over the field of complex numbers and $\sigma\in S_n$. In this paper, we describe the set of eigenvalues of $\rho(\sigma)$. Based on…

A base for a permutation group $G$ acting on a set $\Omega$ is a sequence $\mathcal{B}$ of points of $\Omega$ such that the pointwise stabiliser $G_{\mathcal{B}}$ is trivial. The base size of $G$ is the size of a smallest base for $G$. We…

We study irreducible restrictions from modules over symmetric groups to subgroups. We get reduction results which substantially restrict the classes of subgroups and modules for which this is possible. Such results are known when the…

We study the symmetry properties of autonomous integrating factors from an algebraic point of view. The symmetries are delineated for the resulting integrals treated as equations and symmetries of the integrals treated as functions or…

We classify all the irreducible characters of a symmetric group such that the induced immanant function $d_{\chi}$ vanishes identically on alternate matrices with the entries in the complex field.

For a fixed integer $t \geq 2$, we consider the irreducible characters of representations of the classical groups of types A, B, C and D, namely $\text{GL}_{tn}, \text{SO}_{2tn+1}, \text{Sp}_{2tn}$ and $\text{O}_{2tn}$, evaluated at…

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…

Building on reduction theorems and dimension bounds for symmetric groups obtained in our earlier work, we classify the irreducible restrictions of representations of the symmetric and alternating groups to proper subgroups. Such…

Formulas for the expansion of arbitrary invariant group functions in terms of the characters for the Sp(2N), SO(2N+1), and SO(2N) groups are derived using a combinatorial method. The method is similar to one used by Balantekin to expand…

In this work, we establish a relationship between the sum of irreducible character degrees and the number of twisted involutions associated with the automorphisms of a finite group. We develop algorithmic frameworks for evaluating these…

We give an explicit expression of the normalized characters of the symmetric group in terms of the contents of the partition labelling the representation.

In 1961, Solomon gave upper and lower bounds for the sum of all the entries in the character table of a finite group in terms of elementary properties of the group. In a different direction, we consider the ratio of the character table sum…

Let G_0 be a connected unipotent algebraic group over a finite field F_q, and let G be the unipotent group over an algebraic closure F of F_q obtained from G_0 by extension of scalars. If M is a Frobenius-invariant character sheaf on G, we…

Let $k$ be a field of characteristic two. We prove that a non constant monic polynomial $f\in k[X]$ of degree $n$ is the minimal/characteristic polynomial of a symmetric matrix with entries in $k$ if and only if it is not the product of…

Many eigenvalue matrix models possess a peculiar basis of observables which have explicitly calculable averages. This explicit calculability is a stronger feature than ordinary integrability, just like the cases of quadratic and Coulomb…

We relate the linear asymptotic representation theory of the symmetric groups to its spin counterpart. In particular, we give explicit formulas which express the normalized irreducible spin characters evaluated on a strict partition $\xi$…

In \cite{[CZ]}, Cohen and Zemel showed that for a partition $\lambda \vdash k$, the dimension of the irreducible representation of $S_{n}$ corresponding to the partition $(n-k,\lambda) \vdash n$ is a polynomial of degree $k$ in $n$, whose…

In 1911 Schur computed the spin character values of the symmetric group using two important ingredients: the first one later became famously known as the Schur Q-functions and the second one was certain creative construction of the…

The integral formulae pertaining to the unitary group $\mathsf{U}(d)$ have been comprehensively reviewed, yielding fresh results and innovative proofs. Central to the derivation of these formulae lies the employment of Schur-Weyl duality, a…

Let $X$ be a character table of the symmetric group $S_n$. It is shown that unless $n = 4$ or $n=6$, there is a unique way to assign partitions of $n$ to the rows and columns of $X$ so that for all $\lambda$ and $\nu$, $X_{\lambda\nu}$ is…