Related papers: On the number of non-zero character values in gene…

We study the zero-sharing behavior among irreducible characters of a finite group. For symmetric groups $S_n$, it is proved that, with one exception, any two irreducible characters have at least one common zero. To further explore this…

We prove certain polynomial relations between the values of complex irreducible characters of general finite symmetric groups. We use it to find some sets of conjugacy classes such that no finite symmetric group has a complex irreducible…

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…

Block character of a finite symmetric group S(n) is a positive definite function which depends only on the number of cycles in permutation. We describe the cone of block characters by identifying its extreme rays, and find relations of the…

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ…

The paper studies how to compute irreducible characters of the generalized symmetric group $C_k\wr{S}_n$ by iterative algorithms. After reproving the Ariki-Koike version of the Murnaghan-Nakayama rule by vertex algebraic methods, we…

We prove, for primes $p\ge5$, two inequalities between the fundamental invariants of Brauer $p$-blocks of finite quasi-simple groups: the number of characters in the block, the number of modular characters, the number of height zero…

For a block B of a finite group G there are well-known orthogonality relations for the generalized decomposition numbers. We refine these relations by expressing the generalized decomposition numbers with respect to an integral basis of a…

The covering number of a non-linear character $\chi$ of a finite group $G$ is the least positive integer $k$ such that every irreducible character of $G$ occurs in $\chi^k$. We determine the covering numbers of irreducible characters of the…

In this short note, we classify pairs of conjugacy classes of the symmetric group such that any non-linear irreducible character of the symmetric group vanishes on at least one of them.

We present a strong upper bound on the number k(B) of irreducible characters of a p-block B of a finite group G in terms of local invariants. More precisely, the bound depends on a chosen major B-subsection (u,b), its normalizer N_G(\langle…

A character of a finite group having degree $n$ takes values which may be expressed as sums of $n$ or fewer roots of unity. In this note, we prove a result which describes the irreducible constituents of generalized characters on abelian…

Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $\chi\in\text{Irr}(G)$ such that $\chi(g)=0$ (i.e., $g$…

Let $\chi$ be an irreducible character of a group $G.$ We denote the sum of the codegrees of the irreducible characters of $G$ by $S_c(G)=\sum_{\chi\in {\rm Irr}(G)}{\rm cod}(\chi).$ We consider the question if $S_c(G)\leq S_c(C_n)$ is true…

We consider random character values X(g) of the symmetric group on n symbols, where X is chosen at random from the set of irreducible characters and g is chosen at random from the group, and we show that X(g)=0 with probability tending to…

We provide an example of a finite group with a conjugacy class of average size on which fewer than half of the irreducible characters are either zero or a root of unity.

Let $G$ be a finite group and $N<G$ a normal subgroup with $G/N$ abelian. We show how the conjugacy classes of $G$ in a given coset $qN$ relate to the irreducible characters of $G$ that are not identically $0$ on $qN$. We describe several…

There has been some interest on how the average character degree affects the structure of a finite group. We define, and denote by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ as the number of zeros…

We prove that for any finite group G, the sum across non-identity elements of the squared absolute value of any generalized character of G which does not vanish on all non-identity elements of G is at least |G|/d -1, where d is the maximal…

We call an irreducible character $p$-singular if $p$ divides its degree. We prove a number of equivalent conditions for a character of the symmetric group $S_n$ to be $p$-singular, involving a certain family of conjugacy classes. This…