arXiv subjects

Pavel Turek

Publications and source records attributed to Pavel Turek.

10 recordsLinked to original sources

arXiv2026

New columns in decomposition matrices of symmetric groups for every block

The central unsolved problem in the modular representation theory of symmetric groups is to find the decomposition matrices, which describe how irreducible representations in characteristic zero decompose upon reduction modulo a prime characteristic $p$. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose $p$-divisible hooks have all even arm lengths. In particular in odd characteristic $p$, for every possible block of every possible symmetric group $S_n$, we determine at least one complete column. These columns are multiplicity-free and are described by a recently introduced combinatorial statistic of partitions (depending on $p$), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules $H^{(2^m)}$.

math.RT↗

arXiv2026

Divisible Arm Lengths, Crystal Reflections, and Enumeration of Newly Found Decomposition Columns

In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the present paper we enumerate these partitions and prove that each block of $p$-weight $w$ contains precisely \[ \binom{w+\frac{p-3}{2}}{w} \] such partitions. More generally, for any integers $d,e>1$, we study and enumerate $d$-balanced $e$-regular partitions -- partitions for which every hook of length divisible by $e$ has arm length divisible by $d$. Our first main result is that the crystal (affine) reflections preserve the $d$-balanced property for all $d,e > 1$. It follows that, for fixed $d$, $e$, and $w$, the number of $d$-balanced $e$-regular partitions in a block of $e$-weight $w$ is independent of the $e$-core. We then compute this number by working in RoCK blocks, obtaining an explicit binomial formula valid for every block. We also investigate closely related odd sequences of partitions. Among others, we find the generating function of the number of odd sequences occurring in a block. Alongside their representation-theoretic relevance, we expect these results to be of independent combinatorial interest.

math.CO↗

arXiv2026

Mullineux map: $d$-balanced partitions and $d$-runner matrices

Let $d,e>1$ be two integers. For $e$ prime, the Mullineux map $m_e$ describes tensor products of the irreducible modules of symmetric groups with the sign in characteristic $e$ as well as certain entries of decomposition matrices. Motivated by understanding new columns of decomposition matrices, we prove that if $λ$ is an $e$-regular partition such that $d$ divides the arm length of any rim hook of $λ$ of size divisible by $e$, then $m_e(λ)'$ is a partition such that the arm length of any of its rim hooks of size divisible by $e$ is congruent to $-1$ modulo $d$. We introduce a new parameter for partitions called the $d$-runner matrix and show that if $λ$ is as above, then the $d$-runner matrices of $λ$ and $m_e(λ)'$ agree. This determines $m_e(λ)'$ uniquely. We approach the whole problem combinatorially and take advantage of a new Abacus Mullineux Algorithm introduced in this paper. We also establish equivalent descriptions of the above partitions which provide an alternative version of the main result about the Mullineux map that becomes particularly strong when $d=2$.

math.CO↗

arXiv2025

Multiplicity-free induced characters of symmetric groups

Let $n$ be a non-negative integer. Combining algebraic and combinatorial techniques, we investigate for which pairs $(G,ρ)$ of a subgroup $G$ of the symmetric group $S_n$ and an irreducible character $ρ$ of $G$ the induced character $ρ\!\uparrow^{S_n}$ is multiplicity-free. As a result, for $n\geq 66$, we classify all subgroups $G\leq S_n$ which give rise to such a pair. Moreover, for the majority of these groups $G$ we identify all the possible choices of the irreducible character $ρ$, assuming $n\geq 73$.

math.RT↗

arXiv2025

Proof of the plethystic Murnaghan-Nakayama rule using Loehr's labelled abacus

The plethystic Murnaghan-Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form $p_r\circ h_m$ as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Nicholas A. Loehr. Abacus proofs of Schur function identities. In: SIAM J. Discrete Math. 24.4 (2010), pp. 1356-1370.

math.CO↗

arXiv2025

Symmetric powers of $S^{(n-1,1)}$ and $D^{(n-1,1)}$

Let $p$ be a prime and $n\geq 2$ be a positive integer. We establish new formulae for the decompositions of the first $p-1$ symmetric powers of the Specht module $S^{(n-1,1)}$ and the irreducible module $D^{(n-1,1)}$ in characteristic $p$ as direct sums of Young permutation modules. As an application of the formulae, we show that these symmetric powers have Specht filtration and find the vertices of their indecomposable summands. Our main tool, constructed in this paper, is a lift of a splitting map of a short exact sequence of certain symmetric powers to a splitting map of a short exact sequence of higher symmetric powers. This is a general construction, which can be applied to a broader family of modules.

math.RT↗

arXiv2025

The wreath matrix

Let $k\leq n$ be positive integers and $\mathbb{Z}_{n}$ be the set of integers modulo $n$. A conjecture of Baranyai from 1974 asks for a decomposition of $k$-element subsets of $\mathbb{Z}_{n}$ into particular families of sets called "wreaths". We approach this conjecture from a new algebraic angle by introducing the key object of this paper, the wreath matrix $M$. As our first result, we establish that Baranyai's conjecture is equivalent to the existence of a particular vector in the kernel of $M$. We then employ results from representation theory to study $M$ and its spectrum in detail. In particular, we find all eigenvalues of $M$ and their multiplicities, and identify several families of vectors which lie in the kernel of $M$.

math.CO↗

arXiv2025

Intervals in Dyck paths and the wreath conjecture

Let $ι_{k}(m,l)$ denote the total number of intervals of length $m$ across all Dyck paths of semilength $k$ such that each interval contains precisely $l$ falls. We give the formula for $ι_{k}(m,l)$ and show that $ι_{k}(k,l)=\binom{k}{l}^2$. Motivated by this, we propose two stronger variants of the wreath conjecture due to Baranyai for $n=2k+1$.

math.CO↗

arXiv2024

The representation ring of $\mathrm{SL}_2(\mathbb{F}_p)$ and stable modular plethysms of its natural module in characteristic $p$

Let $p$ be an odd prime and let $k$ be a field of characteristic $p$. We provide a practical algebraic description of the representation ring of $k\mathrm{SL}_2(\mathbb{F}_p)$ modulo projectives. We then investigate a family of modular plethysms of the natural $k\mathrm{SL}_2(\mathbb{F}_p)$-module $E$ of the form $\nabla^ν\mathrm{Sym}^l E$ for a partition $ν$ of size less than $p$ and $0\leq l\leq p-2$. Within this family we classify both the modular plethysms of $E$ which are projective and the modular plethysms of $E$ which have only one non-projective indecomposable summand which is moreover irreducible. We generalise these results to similar classifications where modular plethysms of $E$ are replaced by $k\mathrm{SL}_2(\mathbb{F}_p)$-modules of the form $\nabla^ν V$, where $V$ is a non-projective indecomposable $k\mathrm{SL}_2(\mathbb{F}_p)$-module and $|ν|<p$.

math.RT↗

arXiv2024

Towards odd-sunflowers: temperate families and lightnings

Motivated by odd-sunflowers, introduced recently by Frankl, Pach, and P{á}lv{ö}lgyi, we initiate the study of temperate families: a family $\mathcal{F} \subseteq \mathcal{P}([n])$ is said to be \emph{temperate} if each $A \in \mathcal{F}$ contains at most $|A|$ elements of $\mathcal{F}$ as a proper subset. We show that the maximum size of a temperate family is attained by the middle two layers of the hypercube $\{0,1\}^n$. As a more general result, we obtain that the middle $t+1$ layers of the hypercube maximise the size of a family $\mathcal{F}$ such that each $A \in \mathcal{F}$ contains at most $\sum_{j=1}^t \binom{|A|}{j}$ elements of $\mathcal{F}$ as a proper subset. Moreover, we classify all such families consisting of the maximum number of sets. In the case of intersecting temperate families, we find the maximum size and classify all intersecting temperate families consisting of the maximum number of sets for odd $n$. We also conjecture the maximum size for even $n$.

math.CO↗