[Paper Review] Young module multiplicities and classifying the indecomposable Young permutation modules

Christopher C. Gill|arXiv (Cornell University)|Mar 28, 2012

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper establishes new reduction formulas for $p$-Kostka numbers—the multiplicities of Young modules in Young permutation modules over fields of characteristic $p$. Using the Brauer construction and combinatorics of Young vertices, it proves that multiplying partitions by $p$ preserves these multiplicities, and classifies indecomposable Young permutation modules, particularly for $p=2$, revealing precise conditions under which $M^\lambda$ is indecomposable.

ABSTRACT

We study the multiplicities of Young modules as direct summands of permutation modules on cosets of Young subgroups. Such multiplicities have become known as the p-Kostka numbers. We classify the indecomposable Young permutation modules, and, applying the Brauer construction for p-permutation modules, we give some new reductions for p-Kostka numbers. In particular we prove that p-Kostka numbers are preserved under multiplying partitions by p, and strengthen a known reduction given by Henke, corresponding to adding multiples of a p-power to the first row of a partition.

Motivation & Objective

• To classify all indecomposable Young permutation modules for symmetric groups in positive characteristic.
• To determine new reduction formulas for $p$-Kostka numbers, which count Young module multiplicities in permutation modules.
• To establish that $p$-Kostaka numbers are invariant under multiplication of partitions by $p$, extending known stability results.
• To strengthen existing reductions by showing preservation of $p$-Kostka numbers under adding $p$-power multiples to the first row of partitions.
• To link decomposition numbers of Schur algebras $S(r,r)$ and $S(rp,rp)$ via the new reductions, enabling algorithmic computation.

Proposed method

• Apply the Brauer construction to $p$-permutation modules, using Broué’s framework and Erdmann’s description of Brauer quotients of Young modules.
• Use combinatorial analysis of Young vertices to derive structural properties of $p$-Kostka numbers.
• Leverage Lucas’ Theorem to analyze binomial coefficients modulo $p$, particularly in the context of multiplicity computations.
• Establish a connection between $p$-Kostka numbers and the structure of endomorphism algebras of Young permutation modules.
• Use the theory of $p$-cores and block decomposition to analyze when modules lie outside the principal block.
• Apply the new reductions to derive an algorithm for computing decomposition numbers of $S(r,r)$ from those of $S(rp,rp)$.

Experimental results

Research questions

• RQ1 When is a Young permutation module $M^\lambda$ indecomposable over a field of characteristic $p$?
• RQ2 How do $p$-Kostka numbers behave under multiplication of partitions by $p$?
• RQ3 Can the reduction of $p$-Kostka numbers under adding $p$-powers to the first row of a partition be strengthened?
• RQ4 What is the precise relationship between decomposition numbers of $S(r,r)$ and $S(rp,rp)$?
• RQ5 Which partitions $\lambda$ yield indecomposable $M^\lambda$ when $p=2$ and $r$ is even?

Key findings

• Multiplying partitions by $p$ preserves $p$-Kostka numbers: $[M^{p\lambda}:Y^{p\mu}] = [M^\lambda:Y^\mu]$ for all partitions $\lambda, \mu \vdash r$.
• For $p=2$ and even $r$, the indecomposable Young permutation modules $M^\lambda$ are precisely $M^{(r)}$ and $M^{(r-k_i,k_i)}$ where $k_i$ satisfies $2^{i-1} \leq k_i < 2^i$ and $k_i \equiv \frac{r-2^n}{2} \mod 2^{i-1}$ for $1 \leq i \leq n$, with $2^n \leq r < 2^{n+1}$.
• If $M^\lambda$ is indecomposable, then $M^{2\lambda}$ is also indecomposable, and there is a bijection between indecomposable $M^\lambda$ for $r$ and indecomposable $M^\mu$ for $2r$ excluding $(2r-1,1)$.
• The $p$-Kostka numbers are preserved under adding integer multiples of $p$-powers to the first parts of partitions, strengthening a prior result by Henke.
• When $p=2$ and $r$ is even, $M^{(r-j,j)}$ is indecomposable if and only if $r-2j$ is divisible by $2^{n_j}$, where $n_j$ is the smallest integer with $2^{n_j-1} \leq j < 2^{n_j}$.
• The decomposition numbers of $S(r,r)$ can be algorithmically determined from those of $S(rp,rp)$, particularly for partitions with empty $p$-core.