Interpolation, integrality, and a generalization of Macdonald’s polynomials.

*(English)*Zbl 0861.05063A 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.

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:

05E05 | Symmetric functions and generalizations |