A SUPERCHARACTER TABLE DECOMPOSITION VIA POWER-SUM SYMMETRIC FUNCTIONS
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Abstract
We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over 𝔽 q , into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q -1 ]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group S n given by the transition matrices between Schur functions, monomials and power-sums. We deduce some combinatorial results associated to this decomposition. In particular, we compute the determinant of the supercharacter table.
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