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Interpolation, integrality, and a generalization of Macdonald’s polynomials. (English) Zbl 0861.05063
A partition of length $$\leq n$$ is a vector $$\lambda\in\mathbb{Z}^n_+$$ satisfying $$\lambda_1\geq\cdots\geq \lambda_n\geq0$$, and its weight is $$|\lambda|=\lambda_1+\cdots+\lambda_n$$. The monomial symmetric function $$m_\lambda(x)$$ is the sum $$\sum_\alpha x^{\alpha_1}_1\cdots x^{\alpha_n}_n$$, where $$\alpha$$ ranges over all distinct permutations of $$\lambda$$. The $$m_\lambda$$ form a $$\mathbb{Z}$$-basis for the ring $$\Lambda_n$$ of symmetric integral polynomials in $$x_1,\dots,x_n$$.
I. Macdonald [Symmetric functions and Hall polynomials, 2nd ed. (1995; Zbl 0824.05059)] has defined certain remarkable polynomials $$P_\lambda(x;q,t)$$ in $$\Lambda_n\otimes\mathbb{Q}(q,t)$$ which can be tersely characterized by the following two properties: First, in the expression of $$P_\lambda$$ in terms of symmetric monomials, the coefficient of $$m_\lambda$$ is 1. Second, let $$T_{q,x_i}$$ be the “$$q$$-shift operator” defined by $$T_{q,x_i}f(x_1,\dots,x_n)= f(x_1,\dots,qx_i,\dots,x_n)$$; then $$P_\lambda$$ is an eigenfunction with eigenvalue $$\sum^n_{i=1}q^{\lambda_i}t^{n-i}$$ for the operator $$D$$ defined by $D:=\sum_i A_i(x;t)T_{q,x_i}\quad\text{where } A_i(x;t):= \prod_{j\neq i} {tx_i-x_j\over x_i-x_j}.$ For $$q=0$$ and $$q=t$$, one gets the Hall-Littlewood polynomial $$P_\lambda(x;t)$$ and the Schur polynomial $$s_\lambda$$, respectively, while $$\lim_{t\to1}P_\lambda(x; t^\alpha,t)$$ yields the Jack polynomial $$P^{(\alpha)}_\lambda(x)$$.
Our first result is a generalization of $$P_\lambda(x;q,t)$$ on $$n$$ “$$t$$-parameters.” Thus let $$\tau= (\tau_1,\dots,\tau_n)$$ be indeterminates, and put $$\mathbb{F}=\mathbb{Q}(q,\tau)$$. If $$\mu$$ is a partition, write $$q^{-\mu}\tau$$ for the $$n$$-tuple $$(q^{-\mu_1}\tau_1,\dots,q^{-\mu_n}\tau_n)$$. We show that for each partition $$\lambda$$ of length $$\leq n$$ there is a unique (inhomogeneous) polynomial $$R_\lambda(x;q,\tau)$$ of degree $$|\lambda|$$ in $$\Lambda_n\otimes\mathbb{F}$$ which satisfies: (1) in the expansion of $$R_\lambda$$ in terms of symmetric monomials, the coefficient of $$m_\lambda$$ is 1; (2) $$R_\lambda(q^{-\mu}\tau;q,t)=0$$ for each partition $$\mu\neq\lambda$$ with $$|\mu|\leq|\lambda|$$.
Theorem 1.1 Let $$R_\lambda(x;q,t)$$ be the polynomial obtained from $$R_\lambda(x;q,\tau)$$ by specializing $$\tau_i=t^{-(n-i)}$$; then the top homogeneous component of $$R_\lambda(x;q,t)$$ is $$P_\lambda(x;q,t)$$.
This is proved in Section 3, by showing that $$R_\lambda(x;q,t)$$ is an eigenfunction for a difference operator $$D'$$ closely related to Macdonald’s operator.
Our second result concerns a conjecture of Macdonald about $$P_\lambda(x;q,t)$$. We identify $$\lambda$$ with its diagram consisting of the lattice points $$(i,j)\in\mathbb{Z}^2$$ such that $$1\leq j\leq \lambda_i$$, and $$\lambda'$$ denotes the transposed diagram. For $$s=(i,j)\in\lambda$$ the armlength is $$a(s)=\lambda_i-j$$ and the leglength is $$l(s)=\lambda_j'-i$$, and we put $$c_\lambda(q,t):=\prod_{s\in\lambda}(1- q^{a(s)}t^{l(s)+1})$$. The polynomial $$J_\lambda(x;q,t):=c_\lambda(q,t)P_\lambda(x;q,t)$$ has remarkable integrality properties. Write $$m_d(\lambda)$$ for the number of $$\lambda_i$$’s equal to $$d$$. Let $$S_\lambda(x;t)$$ be the basis dual to $$s_\lambda$$ for the inner product on $$\Lambda_n\otimes\mathbb{Q}(t)$$ defined by $$\langle P_\lambda(x;t), P_\mu(x;t)\rangle=\delta_{\lambda\mu}/\prod_{d\geq 1} \prod^{m_d(\lambda)}_{j=1}(1- t^j)$$. Define the $$(q,t)$$-Kostka coefficients $$K_{\lambda\mu}(q,t)$$ by expressing $$J_\mu(x;q,t):=\sum_\lambda K_{\lambda\mu}(q,t)S_\lambda(x;t)$$. Our second main result, proved in Section 5, is the following.
Theorem 1.2 $$K_{\lambda\mu}(q,t)$$ is a polynomial in $$q$$ and $$t$$ with integral coefficients.

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