2023/07/19 by Munasinghe, Dinushi, Webster, Ben
#FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2307.10406
We study the representation theory of the type B Schur algebra Ln(m) with unequal parameters introduced in work of Lai, Nakano and Xiang. For generic values of q,Q, this algebra is semi-simple and Morita equivalent to the Hecke algebra, but for special values, its category of modules is more complicated. We study this representation theory by comparison with the cyclotomic q-Schur algebra of Dipper, James and Mathas, and use this to construct a cellular algebra structure on Ln(m). This allows us to index the simple Ln(m)-modules as a subset of the set of bipartitions of n. For m large, this will be all bipartitions of n if and only if Ln(m) is quasi-hereditary, in which case, Ln(m) is Morita equivalent to the cyclotomic q-Schur algebra. We prove a modified version of a conjecture of Lai, Nakano and Xiang giving the values of (q,Q) where this holds: if m is large and odd, Q≠ -qk for all k satisfying (4-n)/(2)≤ k