Plethystic formulas for Macdonald \(q,t\)-Kostka coefficients.

*(English)*Zbl 0865.05075In Chapter VI of [M] [Symmetric functions and Hall polynomials (2nd ed. 1995; Zbl 0824.05059)], I. G. Macdonald defines a family of symmetric functions \(P_\mu(x;q,t)\) in a countable sequence of variables \(x=(x_1,x_2,\dots)\) with two parameters \(q\) and \(t\). These functions are parametrised by partitions \(\mu\), and they form a \(F\)-basis of the ring \(\Lambda_F\) of symmetric functions in the \(x_i\) with coefficients in \(F=\text{Q}(q,t)\), the field of rational functions in \(q\) and \(t\) with rational numbers as coefficients. The function \(P_\mu(x;q,t)\) reduces to the Schur function \(s_\mu\) in the case \(q=t\) and to the Hall-Littlewood function \(P_\mu(x;t)\) in the case \(q=0\). Several other important families of symmetric functions can also be obtained as specialisations of \(P_\mu(x;q,t)\).

The transition matrix \(K(t)\) from Schur functions \(s_\lambda(x)\) to Hall-Littlewood functions \(P_\mu(x;t)\) is a strictly upper unitriangular matrix whose entries are polynomials in \(t\) with positive integer coefficients ([M], III.6). For \(t=1\), this is the classical Kostka matrix which enumerates tableaux of given shape and content. Macdonald has conjectured a generalisation of this result for the transition matrices from the functions \(P_\mu(x;q,t)\) to the Schur functions \(S_\lambda(x;t)\). These Schur functions reduce to the classical Schur functions \(s_\mu(x)\) in the case \(t=0\) and can be defined by the formula \(S_\lambda(x;t)=\sum_\rho z^{-1}_\rho\chi^\lambda_\rho p_\rho(x)\prod_i(1-t^{\rho_i})\), where \(p_\rho\) is the usual power sum symmetric function and the integers \(z_\rho\) and \(\chi^\lambda_\rho\) have their usual meaning in the representation theory of the symmetric group.

Macdonald conjectures ([M], VI(8.18)) that the entries of the two-parameter Kostka matrix \(K(q,t)\) given by \(c_\mu(q,t)P_\mu(x;q,t)=\sum_\lambda K_{\lambda,\mu}(q,t)S_\lambda(x;t)\) are polynomials in \(q\) and \(t\) with positive integer coefficients. Here \(c_\mu(q,t)=\prod_s(1-q^{a(s)}t^{l(s)+1})\) is a polynomial defined, in terms of the diagram of \(\lambda\), by the functions which associate to each node \(s\) of the diagram the number of nodes \(a(s)\) in the same row and a higher numbered column, and the number of nodes \(l(s)\) in the same column and a higher numbered row. The matrices \(K_n(q,t)\) (which are not upper triangular) are given for \(n\leq 6\) in ([M], 359-361).

The main result of the present paper proves Macdonald’s conjecture except for the statement that the integer coefficients which appear in the polynomials \(K_{\lambda,\mu}(q,t)\) are positive. This is accomplished by means of an algorithm which yields explicit formulae for the Kostka coefficients \(K_{\lambda,\mu}(q,t)\) in terms of certain symmetric polynomials \(\text{k}_\gamma(x;q,t)\) in \(\Lambda_F\) by a process of ‘plethystic substitution’. This involves substitution in \(\text{k}_\gamma\) for the variables \(x_i\) by appropriate monomials in \(q\) and \(t\) calculated from the diagram of \(\mu\); here \(\gamma\) is the partition obtained by deleting the largest (first) part of \(\lambda\).

The result shows that the Kostka coefficients depend on \(\mu\) in a relatively simple manner. The authors tabulate 12 polynomials \(\text{k}_\gamma\) which suffice to determine the matrices \(K_n(q,t)\) for \(n\leq 8\), with an extended tabulation (in MAPLE input format) available via anonymous FTP from macauley.ucsd.edu. These calculations were carried out using the MAPLE ‘SF’ package of J. Stembridge. This paper contains a wealth of material: applications to the positivity of Macdonald’s conjecture are to follow in a paper by A. M. Garsia and J. Remmel.

The transition matrix \(K(t)\) from Schur functions \(s_\lambda(x)\) to Hall-Littlewood functions \(P_\mu(x;t)\) is a strictly upper unitriangular matrix whose entries are polynomials in \(t\) with positive integer coefficients ([M], III.6). For \(t=1\), this is the classical Kostka matrix which enumerates tableaux of given shape and content. Macdonald has conjectured a generalisation of this result for the transition matrices from the functions \(P_\mu(x;q,t)\) to the Schur functions \(S_\lambda(x;t)\). These Schur functions reduce to the classical Schur functions \(s_\mu(x)\) in the case \(t=0\) and can be defined by the formula \(S_\lambda(x;t)=\sum_\rho z^{-1}_\rho\chi^\lambda_\rho p_\rho(x)\prod_i(1-t^{\rho_i})\), where \(p_\rho\) is the usual power sum symmetric function and the integers \(z_\rho\) and \(\chi^\lambda_\rho\) have their usual meaning in the representation theory of the symmetric group.

Macdonald conjectures ([M], VI(8.18)) that the entries of the two-parameter Kostka matrix \(K(q,t)\) given by \(c_\mu(q,t)P_\mu(x;q,t)=\sum_\lambda K_{\lambda,\mu}(q,t)S_\lambda(x;t)\) are polynomials in \(q\) and \(t\) with positive integer coefficients. Here \(c_\mu(q,t)=\prod_s(1-q^{a(s)}t^{l(s)+1})\) is a polynomial defined, in terms of the diagram of \(\lambda\), by the functions which associate to each node \(s\) of the diagram the number of nodes \(a(s)\) in the same row and a higher numbered column, and the number of nodes \(l(s)\) in the same column and a higher numbered row. The matrices \(K_n(q,t)\) (which are not upper triangular) are given for \(n\leq 6\) in ([M], 359-361).

The main result of the present paper proves Macdonald’s conjecture except for the statement that the integer coefficients which appear in the polynomials \(K_{\lambda,\mu}(q,t)\) are positive. This is accomplished by means of an algorithm which yields explicit formulae for the Kostka coefficients \(K_{\lambda,\mu}(q,t)\) in terms of certain symmetric polynomials \(\text{k}_\gamma(x;q,t)\) in \(\Lambda_F\) by a process of ‘plethystic substitution’. This involves substitution in \(\text{k}_\gamma\) for the variables \(x_i\) by appropriate monomials in \(q\) and \(t\) calculated from the diagram of \(\mu\); here \(\gamma\) is the partition obtained by deleting the largest (first) part of \(\lambda\).

The result shows that the Kostka coefficients depend on \(\mu\) in a relatively simple manner. The authors tabulate 12 polynomials \(\text{k}_\gamma\) which suffice to determine the matrices \(K_n(q,t)\) for \(n\leq 8\), with an extended tabulation (in MAPLE input format) available via anonymous FTP from macauley.ucsd.edu. These calculations were carried out using the MAPLE ‘SF’ package of J. Stembridge. This paper contains a wealth of material: applications to the positivity of Macdonald’s conjecture are to follow in a paper by A. M. Garsia and J. Remmel.

Reviewer: G.Walker (Manchester)

##### MSC:

05E05 | Symmetric functions and generalizations |