arXiv subjects
Publications and source records attributed to Jorn B. Olsson.
In this article we consider hook removal operators on odd partitions, i.e., partitions labelling odd-degree irreducible characters of finite symmetric groups. In particular we complete the discussion, started by Isaacs, Navarro, Olsson and Tiep in 2016, concerning the commutativity of such operators.
Let $n$ and $k$ be natural numbers such that $2^k < n$. We study the restriction to $\mathfrak{S}_{n-2^k}$ of odd-degree irreducible characters of the symmetric group $\mathfrak{S}_n$. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sem. Lothar. Combin. 75 (2015), Art. B75g, 13 pages] and recently developed in [Isaacs I.M., Navarro G., Olsson J.B., Tiep P.H., J. Algebra 478 (2017), 271-282].
In this paper, we present, given a odd integer $d$, a decomposition of the multiset of bar lengths of a bar partition $λ$ as the union of two multisets, one consisting of the bar lengths in its $\bar{d}$-core partition $\bar{c}_d(λ)$ and the other consisting of modified bar lengths in its $\bar{d}$-quotient partition. In particular, we obtain that the multiset of bar lengths in $\bar{c}_d(λ)$ is a sub-multiset of the multiset of bar lengths in $λ$. Also we obtain a relative bar formula for the degrees of spin characters of the Schur extensions of the symmetric group. The proof involves a recent similar result for partitions, proved in [1].
In this paper, we present, for any integer d, a description of the set of hooks in a d-symbol. We then introduce generalized hook length functions for a d-symbol, and prove a general result about them, involving the core and quotient of the symbol. We list some applications, for example to the well-known hook lengths in integer partitions. This leads in particular to a generalization of a relative hook formula for the degree of characters of the symmetric group discovered by G. Malle and G. Navarro in [3].
A special type of conjugacy classes in symmetric groups is studied and used to answer a question about odd-degree irreducible characters
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, based on the theory of Hall-Littlewood symmetric functions, of the determinant of the regular character table of S_n with respect to an integer r>1. This result had earlier been proved by Olsson in a longer and more indirect manner. As a consequence, we obtain a new proof of the Mathas' Conjecture on the determinant of the Cartan matrix of the Iwahori-Hecke algebra. When r is prime we determine the Smith normal form of the regular character table. Taking r large yields the Smith normal form of the full character table of S_n. Analogous results are then given for spin characters.