Defect 2 spin blocks of symmetric groups and canonical basis coefficients

Fayers, Matthew

doi:10.1090/ert/600

Defect $2$ spin blocks of symmetric groups and canonical basis coefficients
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by Matthew Fayers

Represent. Theory 26 (2022), 134-178

DOI: https://doi.org/10.1090/ert/600

Published electronically: March 17, 2022

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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 $A^{(2)}_{2n}$.

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Representation Theory

Defect $2$ spin blocks of symmetric groups and canonical basis coefficients
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Abstract: