[Paper Review] On signed p-Kostka numbers and the indecomposable signed Young permutation modules

Eugenio Giannelli, Kay Jin Lim+2 more|arXiv (Cornell University)|May 5, 2015

Algebraic structures and combinatorial models9 references3 citations

TL;DR

This paper establishes the existence and main properties of signed Young modules for the symmetric group using only basic symmetric group representation theory and the Broué correspondence, proving new reduction theorems for signed $p$-Kostka numbers and fully classifying indecomposable signed Young permutation modules and their endomorphism algebras.

ABSTRACT

We prove the existence and main properties of signed Young modules for the symmetric group, using only basic facts about symmetric group representations and the Brou{é} correspondence. We then prove new reduction theorems for the signed $p$-Kostka numbers, defined to be the multiplicities of signed Young modules as direct summands of signed Young permutation modules. We end by classifying the indecomposable signed Young permutation modules and determining their endomorphism algebras.

Motivation & Objective

• To provide an independent, self-contained proof of the existence and properties of signed Young modules using only basic symmetric group representation theory and the Broué correspondence, avoiding the Schur superalgebra.
• To establish new reduction theorems for signed $p$-Kostka numbers, which count the multiplicities of signed Young modules in signed Young permutation modules.
• To classify all indecomposable signed Young permutation modules and determine their endomorphism algebras, completing the structural understanding of these modules in positive characteristic.
• To clarify the relationship between signed Young modules and Specht modules, particularly in light of Hemmer's conjecture on self-dual modules with Specht filtrations, which was later shown to be false.
• To unify and extend previous results on $p$-Kostka numbers by generalizing them to the signed setting and providing a framework for computing decomposition matrices in characteristic $p$.

Proposed method

• Uses the Broué correspondence for $p$-permutation modules to relate the structure of signed Young permutation modules to their vertices and sources.
• Applies the $p$-adic expansion of partitions to define the vertices of signed Young modules as Sylow $p$-subgroups of appropriate Young subgroups.
• Employs the Littlewood–Richardson rule and Specht filtration techniques to analyze the structure of induced modules and their decompositions.
• Utilizes the isomorphism between the $P$-fixed points of signed Young permutation modules and those of signed Young modules to compare their structures via the Broué correspondence.
• Applies the canonical surjection from wreath product groups to define the modules $W_{p^i}((m_i)|(n_i))$ and $Q_{p^i}((m_i)|(n_i))$, which are shown to be isomorphic in the relevant cases.
• Leverages the fact that for $m_i, n_i < p$, the projective covers of trivial modules are trivial, allowing simplification of induced modules to match the corresponding signed Young module components.

Experimental results

Research questions

• RQ1 Can signed Young modules be constructed independently of the Schur superalgebra using only basic symmetric group representation theory?
• RQ2 What reduction theorems govern the behavior of signed $p$-Kostka numbers under $p$-fold scaling of partitions?
• RQ3 Which signed Young permutation modules are indecomposable, and what are their endomorphism algebras?
• RQ4 How do the labels of indecomposable signed Young permutation modules relate to the $p$-adic expansions of their partition parameters?
• RQ5 To what extent do signed Young modules coincide with self-dual modules having Specht filtrations, and how do they relate to simple Specht modules?

Key findings

• The paper proves that signed Young modules $Y(\lambda|p\mu)$ exist and are uniquely characterized as the only summands of $M(\alpha|\beta)$ that appear in $M(\alpha'|\beta')$ only when $(\lambda|p\mu) \unrhd (\alpha'|\beta')$, confirming their uniqueness and structure.
• A new reduction theorem is established: $[M(p\alpha|p\beta):Y(p\lambda|p^2\mu)] \leq [M(\alpha|\beta):Y(\lambda|p\mu)]$, with equality if certain $\delta_0$ conditions vanish, showing a strong control on the growth of signed $p$-Kostka numbers.
• The indecomposable signed Young permutation modules are fully classified: $M((m)|\varnothing) \cong Y((m)|\varnothing)$, $M(\varnothing|(n)) \cong Y((1^{n_0})|(pn'))$, and if $m+n \equiv 0 \pmod{p}$, then $M((m)|(n)) \cong Y((m,1^{n_0})|(pn'))$.
• The endomorphism algebra of $M((m)|(n))$ is shown to be isomorphic to the endomorphism algebra of $Y((m,1^{n_0})|(pn'))$, and for $M((kp-1,1)|(m))$ with $m \geq 2$, the module is decomposable due to block separation of Specht factors.
• For $p$-adic components with $m_i + n_i = p$, the module $W_1((m_0)|(n_0))$ is shown to be isomorphic to the signed Young module $Y((m_0,1^{n_0})|\varnothing)$, confirming the compatibility of the Broué correspondence in the critical case.
• The proof shows that $W_{p^i}((m_i)|(n_i)) \cong Q_{p^i}((m_i)|(n_i))$ for $m_i, n_i < p$, which is essential in matching the $P$-fixed point modules of the signed Young permutation and signed Young modules, thus proving the isomorphism of the original modules.