Bessenrodt passed away at the age of 63 after a short serious illness.Mathematics has lost an original, dedicated, and prolific scientist and teacher of international class, who will be sorely missed.Christine will be remembered for her unwavering devotion to all of her professional activities, both scientific and administrative.

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].

Let n and k be natural numbers such that 2 < n. We study the restriction to Sn−2k of odd-degree irreducible characters of the symmetric group Sn. This analysis completes the study begun in [Ayyer A., Prasad A., Spallone S., Sém. 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].

We prove that the double covers of the alternating and symmetric groups are determined by their complex group algebras. To be more precise, let n >= 5 be an integer, G a finite group, and let (A) over cap (n) and (S) over cap (n) (+/-) denote the double covers of A(n) and S-n, respectively. We prove that CG congruent to C (A) over cap (n) if and only if G congruent to (A) over cap (n), and CG congruent to C (S) over cap (+)(n) congruent to (S) over cap (-)(n) if and only if G congruent to (S) over cap (+)(n) or (S) over cap (-)(n). This in particular completes the proof of a conjecture proposed by the second and fourth authors that every finite quasisimple group is determined uniquely up to isomorphism by the structure of its complex group algebra. The known results on prime power degrees and relatively small degrees of irreducible (linear and projective) representations of the symmetric and alternating groups together with the classification of finite simple groups play an essential role in the proofs.

We give a complete classification of the unique path partitions and study congruence properties of the function which enumerates such partitions.

AbstractWe present, given an odd integerd, 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 itsd-core partitioncd(λ) and the other consisting of modified bar lengths in itsd-quotient partition. In particular, we obtain that the multiset of bar lengths incd(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$\mathfrak{S}_n$. The proof involves a recent similar result for partitions, proved by Bessenrodt and the authors.

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on restricted sets of conjugacy classes, like beautiful combinatorial determinant formulae for submatrices of the character table and Cartan matrices with respect to basic sets; we observe that similar phenomena occur for the transition matrices between power sum symmetric functions to bounded partitions and the k-Schur functions defined by Lapointe and Morse. Arithmetic properties of the numbers occurring in this context are studied via generating functions.

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 Trans. Am. Math. Soc. 363, 6647–6669, 2011.

SummaryIn the June 2012 issue of this Magazine, Frumosu and Teodorescu-Frumosu proved that, for all integers m ≥ 2, where the inner sum is taken over all p-term ordered partitions of m. Their proof is calculus-based, relying on power series manipulations. In this note, we provide a combinatorial proof of this identity (which they requested at the end of their article) and we use the insights gained via this argument to prove other results of a similar type.

If s and t are relatively prime positive integers we show that the s -core of a t -core partition is again a t -core partition. A similar result is proved for bar partitions under the additional assumption that s and t are both odd.

A special type of conjugacy classes in symmetric groups is studied and used to answer a question about odd-degree irreducible characters.

We prove some new results on core partitions and apply them to describe explicitly all block coverings in symmetric groups.

Necessary and sufficient conditions are given for an s-block of integer partitions to be contained in a t-block. The generating function for such partitions is found analytically, and also bijectively, using the notion of an (s, t)-abacus. The largest partition which is both an s-core and a t-core is explicitly given.

We investigate the problem of finding a set of prime divisors of the order of a finite group, such that no two irreducible characters are in the same p‐block for all primes p in the set. Our main focus is on the simple and quasi‐simple groups. For results on the alternating and symmetric groups and their double covers, some combinatorial results on the cores of partitions are proved, which may be of independent interest. We also study the problem for groups of Lie type. The sporadic groups (and their relatives) are checked using GAP.

We prove that a set of characters of a finite group can only be the set of characters for principal blocks of the group at two different primes when the primes do not divide the group order. This confirms a conjecture of Navarro and Willems in the case of principal blocks.

We determine the elementary divisors of the Cartan matrices of spin p p -blocks of the covering groups of the symmetric groups when p p is an odd prime. As a consequence, we also compute the determinants of these Cartan matrices, and in particular we confirm a conjecture by Brundan and Kleshchev that these determinants depend only on the weight but not on the sign of the block.

In this paper we classify all block equalities and all nontrivial block inclusions for spin blocks of the double covers of the symmetric groups at different odd primes. More generally, we describe for an odd integer s>1 explicitly when an s¯-block of bar partitions is contained in a t¯-block of bar partitions, for t>1 an odd integer or t=4. The question of block equality leads to the study of (s¯,t¯)-cores. In the case of primes they label spin characters which are of defect 0 for different primes and therefore represent a block equality. We enumerate these cores and show that there is a unique maximal one.