Related papers: Symmetric Group Character Degrees and Hook Numbers

A basic problem for any class of nonassociative algebras is to determine the polynomial identities satisfied by the symmetrization and the skew-symmetrization of the original product. We consider the symmetrization of the product in the…

We develop a theory of levels for irreducible representations of symmetric groups of degree $n$ analogous to the theory of levels for finite classical groups. A key property of level is that the level of a character, provided it is not too…

If a group $G$ is $\pi$-separable, where $\pi$ is a set of primes, the set of irreducible characters $\operatorname{B}_{\pi}(G) \cup \operatorname{B}_{\pi'}(G)$ can be defined. In this paper, we prove that there are variants of some…

We show how infinite series of a certain type involving generalized harmonic numbers can be computed using a knowledge of symmetric functions and multiple zeta values. In particular, we prove and generalize some identities recently…

Jack characters are a one-parameter deformation of the characters of the symmetric groups; a deformation given by the coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions. We study Jack…

We show that for any knot there exist only finitely many irreducible metabelian characters in the $SL(2,\mathbb{C})$-character variety of the knot group, and the number is given explicitly by using the determinant of the knot. Then it turns…

Let $S$ be a Suzuki group $^2B_2(q^2)$, where $q^2=2^{2f+1}$, $f\geqslant 1$. In this paper, we determine the degrees of the ordinary complex irreducible characters of every group $G$ such that $S\leqslant G\leqslant \Aut(S)$.

Let p be an odd prime, and A_n the alternating group of degree n. We determine which ordinary irreducible representations of A_n remain irreducible in characteristic p, verifying the author's conjecture from [Represent. Theory 14, 601-626].…

Let $k$ be a positive integer. Let $G$ be a balanced bipartite graph of order $2n$ with bipartition $(X, Y)$, and $S$ a subset of $X$. Suppose that every pair of nonadjacent vertices $(x,y)$ with $x\in S, y\in Y$ satisfies $d(x)+d(y)\geq…

We determine invariants like the Smith normal form and the determinant for certain integral matrices which arise from the character tables of the symmetric groups S_n and their double covers. In particular, we give a simple computation,…

We use homotopy theory to define certain rational coefficients characteristic numbers with integral values, depending on a given prime number q and positive integer t. We prove the first nontrivial degree formula and use it to show that…

In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable…

There exists a biderivation structure on the polynomial algebra $\mathscr{A}[n] = K[x_1,\dots,x_n],$ where $K$ is a field with $\operatorname{char}(K)\ne 2$, defined by $f \circ h = \sum_{i=1}^n \frac{\partial f}{\partial…

In an earlier paper [1] it was shown that the Frobenius compound characters for the symmetric groups are related to the irreducible characters by a linear relation that involves a unitriagular coupling matrix that gives the Frobenius…

In this paper, we prove the existence of a relation between the prime divisors of the order of a Sylow normalizer and the degree of characters having values in some small cyclotomic fields. This relation is stronger when the group is…

Linear recursions of degree $k$ are determined by evaluating the sequence of Generalized Fibonacci Polynomials, $\{F_{k,n}(t_1,...,t_k)\}$ (isobaric reflects of the complete symmetric polynomials) at the integer vectors $(t_1,...,t_k)$. If…

Let $U$ be a maximal unipotent subgroup in a symplectic group over a finite field of sufficiently large characteristic $p$. According to the Kirillov's orbit method, the coadjoint orbits of the group $U$ play the key role in the description…

This note will give an enumeration of $n$-cycles in the symmetric group ${\mathcal S}_n$ by their degree (also known as their cyclic descent number) and studies similar counting problems for the conjugacy classes of $n$-cycles under the…

We prove that the number of irreducible ordinary characters in the principal $p$-block of a finite group $G$ of order divisible by $p$ is always at least $2\sqrt{p-1}$. This confirms a conjecture of H\'{e}thelyi and K\"{u}lshammer for…

The degree chromatic polynomial $Pm(G,k)$ of a graph $G$ counts the number of $k$-colorings in which no vertex has $m$ adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree…