In this note, we prove a recent conjecture of Hung, Martínez and Navarro on the sums of character degrees of finite groups.

Let P_n be a Sylow p-subgroup of the symmetric group S_n. We investigate the number and sizes of the P_n∖ S_n / P_n double cosets, showing that most double cosets have maximal size when p is odd, or equivalently, that P_n∩ P_n^x=1 for most x∈ S_n when n is large. We also find that all possible sizes of such double cosets occur, modulo a list of small exceptions.

We show that the p-part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of p-power order. As a corollary we deduce that the set of zeros of prime power order controls the degree of such a character. The same problem is analysed for alternating groups, where we show that when p=2 this data can only be determined up to two possibilities. We prove analogous statements for the defect of the p-block containing the character and for the p-height of the character.

We show that the p $p$ -part of the degree of an irreducible character of a symmetric group is completely determined by the set of vanishing elements of p $p$ -power order. As a corollary, we deduce that the set of zeros of prime power order controls the degree of such a character. The same problem is analysed for alternating groups, where we show that when p = 2 $p=2$ these data can only be determined up to two possibilities. We prove analogous statements for the defect of the p $p$ -block containing the character and for the p $p$ -height of the character.

. Let p >= 5 be a prime and let P be a Sylow p-subgroup of a finite symmetric group Sn. To every irreducible character of P we associate a collection of labelled, complete p-ary trees. The main results of this article describe the positivity of Sylow branching coefficients for all irreducible characters of P in terms of combinatorial properties of these trees, extending previous work on the linear characters of P.

We prove a recursive formula for plethysm coefficients of the form $a^\mu_{\lambda,(m)}$, generalising results on plethysms due to Bruns--Conca--Varbaro and de Boeck--Paget--Wildon. From this we deduce a stability result and resolve two conjectures of de Boeck concerning plethysms, as well as obtain new results on Sylow branching coefficients for symmetric groups for the prime 2. Further, letting $P_n$ denote a Sylow 2-subgroup of $S_n$, we show that almost all Sylow branching coefficients of $S_n$ corresponding to the trivial character of $P_n$ are positive.

In this note, we prove some new stability results for plethysm coefficients. As special cases, we verify a conjecture of Wildon, and show the stability of sequences recently predicted by Bessenrodt, Bowman and Paget [Trans. Amer. Math. Soc. 375 (2022), pp. 5151-5194] to be weakly increasing.

In this article we investigate the positivity of Sylow branching coefficients for symmetric groups when p=3. In particular, we complete the discussion begun by Giannelli and the author in Giannelli and Law (2018) [4] and developed in Giannelli and Law (2021) [5] concerning the case of odd primes.

We prove that a finite group G$G$ has a normal Sylow p$p$ ‐subgroup P$P$ if, and only if, every irreducible character of G$G$ appearing in the permutation character (1P)G$({\bf 1}_P)^G$ with multiplicity coprime to p$p$ has degree coprime to p$p$ . This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.

We prove that a finite group G $G$ has a normal Sylow p $p$ -subgroup P $P$ if, and only if, every irreducible character of G $G$ appearing in the permutation character ( 1 P ) G $({\bf 1}_P)^G$ with multiplicity coprime to p $p$ has degree coprime to p $p$ . This confirms a prediction by Malle and Navarro from 2012. Our proof of the above result depends on a reduction to simple groups and ultimately on a combinatorial analysis of the properties of Sylow branching coefficients for symmetric groups.

Let A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A –modules P into those of the torsion submodules of P . As an application, we show that blocks of both the classical and quantum Schur algebras S (2, r ) and S q (2, r ) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2 p k simple modules for some k .

Let $p$ be a prime number. In this article we study the restriction to $\mathfrak{S}_{n-1}$ of irreducible characters of degree coprime to $p$ of $\mathfrak{S}_n$. In particular, we study the combinatorial properties of the subgraph $\mathbb{Y}_{p'}$ of the Young graph $\mathbb{Y}$. This is an extension to odd primes of the work done by Ayyer, Prasad and Spallone for $p=2$.

We suppose throughout that G is a finite group with a faithful matrix representation X over the complex field. We suppose that X affords a character π of degree r whose values are rational (hence rational integers). If the matrices in some representation of G affording a character π0 are all permutation matrices, then π0 is called a permutation character. Permutation characters have non-negative integral values. In the general case, we consider what properties of permutation characters are true of π, and in particular, under what circumstances π is a permutation character. Note that assuming X to b faithful is equivalent to considering the image group X(G) instead of G.