A Relationship between the Major Index for Tableaux and the Charge Statistic for Permutations

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 Sλ q of 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 Mλ q 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(Mλ q ) = ΣμKμλdim(S μ q ), 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) − fλ(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, SY T (λ), to Wλ such that maj(T ) = ch(f(T )) for T ∈ SY T (λ).