A relationship between the major index for tableaux and the charge statistic for permutations

Abstract

The widely studied q-polynomial fλ(q), which specializes when q = 1 to fλ, the number of standard Young tableaux of shape λ, has multiple combinatorial interpretations. It represents the dimension of the unipotent representation Sqλof the finite general linear group GLn(q), it occurs as a special case of the Kostka-Foulkes polynomials, and it gives the generating function for the major index statistic on standard Young tableaux. Similarly, the q-polynomial gλ(q) has combinatorial interpretations as the q-multinomial coefficient, as the dimension of the permutation representation M qλ of the general linear group GLn(q), and as the generating function for both the inversion statistic and the charge statistic on permutations in Wλ. It is a well known result that for λ a partition of n, dim(Mqλ) = ∑μKμλdim(Sqμ), where the sum is over all partitions μ of n and where the Kostka number Kμλ gives the number of semistandard Young tableaux of shape μ and content λ. Thus gλ(q) - f λ(q) is a q-polynomial with nonnegative coefficients. This paper gives a combinatorial proof of this result by defining an injection f from the set of standard Young tableaux of shape λ, SYT(λ), to W λ such that maj(T) = ch(f(T)) for T ∈ SYT(λ).