On the image of the trivial source ring in the ring of virtual characters of a finite group

The Murnaghan-Nakayama rule and some virtual S_n characters

We construct certain virtual characters for the symmetric groups, then compute a formula which calculates the values of these virtual characters.

Trivial source character tables of $\text{SL}_2(q)$, Part II

We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups $\text{SL}_{2}(q)$ for $q$ even, over a large enough field $k$ of positive characteristic ${\ell}$ not dividing $q$. This article is a continuation of our article Trivial Source Character Tables of $\text{SL}_{2}(q)$ where we considered, in particular, the case in which $q$ is odd in cross-characteristic.

Trivial source character tables of SL(2,q)

We compute the trivial source character tables (also called species tables of the trivial source ring) of the infinite family of finite groups SL(2,q) over a large enough field of positive characteristic $\ell$ via character-theoretical methods in the cases in which $q$ is odd, $\ell \mid (q\pm1)$ when~$\ell$ is odd, and $q\equiv \pm 3\pmod{8}$ when $\ell=2$.

A virtual character and nonzero Kronecker coefficients for self-conjugate partitions

We generalize Regev's result on a virtual character of $S_n$. Suppose that $λ$ and $μ$ are integer partitions of $n$. For the associated irreducible character $χ^λ$ of $S_n$, when $χ^λ(μ)\neq0$ we find another partition $τ$ related to $λ$ such that $χ^τ(μ)\neq0$ by the virtual character. Applying this result, we obtain a class of nonzero Kronecker coefficients by Pak et al.'s character criterion. Moreover, we discuss the effectiveness of Pak et al.'s character criterion by a concrete example.

The orthogonal unit group of the trivial source ring

Let $G$ be a finite group, $p$ a prime, and $(K,\mathcal{O},F)$ a $p$-modular system. We prove that the trivial source ring of $\mathcal{O} G$ is isomorphic to the ring of {\em coherent} $G$-stable tuples $(χ_P)$, where $χ_P$ is a virtual character of $K[N_G(P)/P]$, $P$ runs through all $p$-subgroups of $G$, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of $G$ with the group of coherent $G$-stable tuples $(…

特征栈的虚类

Virtual Classes of Character Stacks

本文将计算闭可定向曲面G-表示簇虚类的拓扑量子场论推广到特征栈情形，定义可表栈格罗滕迪克环，得到特征栈虚类的动机分解，并在两类群中验证了方法的有效性与优势。