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Plethystic formulas for Macdonald $$q,t$$-Kostka coefficients. (English) Zbl 0865.05075
In 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.

##### MSC:
 500000 Symmetric functions and generalizations
Maple
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