Defect 2 spin blocks of symmetric groups and canonical basis coefficients

2019/05/10 by Matthew Fayers, Fayers, Matthew
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1905.04080

Abstract

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect 2, analogous to Richards's formula for defect 2 blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding "q-decomposition numbers", i.e. the canonical basis coefficients in the level-1 q-deformed Fock space of type A⁽²⁾_2n; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic 2n+1. Along the way, we prove some general results on q-decomposition numbers. This paper represents the first substantial progress on canonical bases in type A⁽²⁾_2n.

Related

• Generalised row and column removal results for decomposition numbers of spin representations of symmetric groups in characteristic 2
• Decomposition numbers of 2-parts spin representations of symmetric groups in characteristic 2
• Symmetric group blocks of small defect
• Composition factors of 2-parts spin representations of symmetric groups
• Comparing Fock spaces in types A⁽¹⁾ and A⁽²⁾