# Determinant connection between Schur polynomials and power sum polynomials

Let $$f_i=f_i(x_1,x_2,\ldots, x_n),i=0,1,2, \ldots$$ be a family of symmetric polynomials. For the partition $$\lambda=(\lambda_1,\lambda_2, \ldots, \lambda_n)$$ consider the determinant $$D_\lambda(f)=\left | \begin{array}{lllll} f_{\lambda_1} & f_{\lambda_1+1} & f_{\lambda_1+2} & \ldots & f_{\lambda_1+n-1}\\ f_{\lambda_2-1} & f_{\lambda_2} & f_{\lambda_2+1} & \ldots & f_{\lambda_2+n-2}\\ \vdots & \vdots & \vdots & \ldots & \vdots \\ f_{\lambda_n-(n-1)} & f_{\lambda_n-(n-2)} & f_{\lambda_n-(n-3)} & \ldots & f_{\lambda_n} \end{array} \right|.$$ It is well known (Jacobi−Trudi formulas) that for the elementary symmetric polynomials $$e_i=e_i(x_1,x_2,\ldots, x_n)$$ and for the complete homogeneous symmetric polynomials $$h_i=h_i(x_1,x_2,\ldots, x_n)$$ we have $$D_\lambda(h)=s_{\lambda}(x_1,x_2,\ldots, x_n) \text{ and } D_\lambda(e)=s_{\lambda'}(x_1,x_2,\ldots, x_n),$$ where $$s_{\lambda}(x_1,x_2,\ldots, x_n)$$ is the Schur polynomial and $$\lambda'$$ is the conjugate partition.

Question. Is there a similar expression for $$D_\lambda(p)$$ where $$p_i=p_i(x_1,x_2,\ldots, x_n)$$ is the power sum symmetric polynomials?

By direct calculation for $$n=2, \lambda_2\geq 1$$ I got that

$$D_{(\lambda_1,\lambda_2)}(p)=-s_{(\lambda_1-1,\lambda_2-1)} V(x_1,x_2)^2$$ and for $$n=3,\lambda_3\geq 2$$

$$D_{(\lambda_1,\lambda_2,\lambda_3)}(p)=-\frac{s_{(\lambda_1-1,\lambda_2-1,\lambda_3-1)}}{s_{(1,1,1)}} V(x_1,x_2,x_3)^2$$ and so on.

Here $$V$$ is the Vandermonde determinant.

I hope that must be a nice formula and for any $$n$$.

I don't know a fully general result, but your pattern for partitions $$\lambda$$ of length $$\leq n$$ with $$n$$-th entry $$\lambda_{n}\geq n-1$$ and with $$n$$ indeterminates persists:

Theorem 1. Let $$n$$ be a positive integer. Let $$\lambda=\left( \lambda _{1},\lambda_{2},\ldots,\lambda_{n}\right)$$ be an integer partition with at most $$n$$ parts. Assume that $$\lambda_{n}\geq n-1$$. Consider polynomials in $$n$$ indeterminates $$x_{1},x_{2},\ldots,x_{n}$$. For each nonnegative integer $$k$$, we set \begin{align*} p_{k}:=x_{1}^{k}+x_{2}^{k}+\cdots+x_{n}^{k}. \end{align*} (This is the $$k$$-th power-sum symmetric polynomial in $$x_{1},x_{2} ,\ldots,x_{n}$$ when $$k>0$$. We have $$p_{0}=n$$.) Define the $$n\times n$$-matrix \begin{align*} P:=\left( p_{\lambda_{i}-i+j}\right) _{1\leq i\leq n,\ 1\leq j\leq n}=\left( \begin{array} [c]{cccc} p_{\lambda_{1}} & p_{\lambda_{1}+1} & \cdots & p_{\lambda_{1}+n-1}\\ p_{\lambda_{2}-1} & p_{\lambda_{2}} & \cdots & p_{\lambda_{2}+n-2}\\ \vdots & \vdots & \ddots & \vdots\\ p_{\lambda_{n}-n+1} & p_{\lambda_{n}-n+2} & \cdots & p_{\lambda_{n}} \end{array} \right) . \end{align*} Let $$\mu=\left( \mu_{1},\mu_{2},\ldots,\mu_{n}\right)$$ be the partition defined by \begin{align*} \mu_{i}=\lambda_{i}-\left( n-1\right) \ \ \ \ \ \ \ \ \ \ \text{for each }i\in\left\{ 1,2,\ldots,n\right\} . \end{align*} (This is indeed a partition, since $$\mu_{n}=\underbrace{\lambda_{n}}_{\geq n-1}-\left( n-1\right) \geq0$$.) Let $$s_{\mu}$$ be the corresponding Schur polynomial in the $$n$$ indeterminates $$x_{1},x_{2},\ldots,x_{n}$$. Furthermore, let \begin{align*} V_{n}:=\prod_{1\leq i be the Vandermonde determinant. Then, \begin{align*} \det P=\left( -1\right) ^{n\left( n-1\right) /2}s_{\mu}\cdot V_{n}^{2}. \end{align*}

Proof. Let $$A_{\mu}$$ be the $$n\times n$$-matrix \begin{align*} \left( x_{j}^{\mu_{i}+n-i}\right) _{1\leq i\leq n,\ 1\leq j\leq n}=\left( \begin{array} [c]{cccc} x_{1}^{\mu_{1}+n-1} & x_{2}^{\mu_{1}+n-1} & \cdots & x_{n}^{\mu_{1}+n-1}\\ x_{1}^{\mu_{2}+n-2} & x_{2}^{\mu_{2}+n-2} & \cdots & x_{n}^{\mu_{2}+n-2}\\ \vdots & \vdots & \ddots & \vdots\\ x_{1}^{\mu_{n}+n-n} & x_{2}^{\mu_{n}+n-n} & \cdots & x_{n}^{\mu_{n}+n-n} \end{array} \right) . \end{align*} It is then well-known that $$$$s_{\mu}=\dfrac{\det\left( A_{\mu}\right) }{V_{n}} . \label{darij1.eq.slam=frac} \tag{1}$$$$ Indeed, this is the alternant formula for Schur polynomials. For a proof, see, e.g., Corollary 2.6.7 in the lecture notes Darij Grinberg and Victor Reiner, Hopf Algebras in Combinatorics, arXiv:1409.8356v7. (The notations in those notes are not quite ours. Namely, our matrix $$A_{\mu}$$ is the transpose of the matrix whose determinant is $$a_{\mu+\rho}$$ in the notes, whereas our $$V_{n}$$ is $$a_{\rho}$$ in these notes. Corollary 2.6.7 has to be applied to $$\mu$$ instead of $$\lambda$$.)

Let $$B$$ be the $$n\times n$$-matrix \begin{align*} \left( x_{i}^{j-1}\right) _{1\leq i\leq n,\ 1\leq j\leq n}=\left( \begin{array} [c]{cccc} 1 & x_{1} & \cdots & x_{1}^{n-1}\\ 1 & x_{2} & \cdots & x_{2}^{n-1}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & x_{n} & \cdots & x_{n}^{n-1} \end{array} \right) . \end{align*} The Vandermonde determinant formula yields \begin{align*} \det B & =\prod_{1\leq i

However, we have $$$$A_{\mu}B=P. \label{darij1.eq.AB=P} \tag{2}$$$$ (Indeed, for any $$i,j\in\left\{ 1,2,\ldots,n\right\}$$, the $$\left( i,j\right)$$-th entry of the matrix $$A_{\mu}B$$ is \begin{align*} \sum_{k=1}^{n}\underbrace{x_{k}^{\mu_{i}+n-i}x_{k}^{j-1}}_{\substack{=x_{k} ^{\mu_{i}+n-i+j-1}=x_{k}^{\lambda_{i}-i+j}\\\text{(since }\mu_{i}=\lambda _{i}-\left( n-1\right) \text{ and}\\\text{thus }\mu_{i}+n-i+j-1=\lambda _{i}-\left( n-1\right) +n-i+j-1=\lambda_{i}-i+j\text{)}}} & =\sum_{k=1} ^{n}x_{k}^{\lambda_{i}-i+j}\\ & =x_{1}^{\lambda_{i}-i+j}+x_{2}^{\lambda_{i}-i+j}+\cdots+x_{n}^{\lambda _{i}-i+j}=p_{\lambda_{i}-i+j}, \end{align*} which happens to be precisely the $$\left( i,j\right)$$-th entry of the matrix $$P$$. Thus, \eqref{darij1.eq.AB=P} follows.)

Now, the two matrices $$A_{\mu}$$ and $$B$$ are square matrices. Hence, \begin{align*} \det\left( A_{\mu}B\right) & =\underbrace{\det\left( A_{\mu}\right) }_{\substack{=s_{\mu}V_{n}\\\text{(by \eqref{darij1.eq.slam=frac})}} }\cdot\underbrace{\det B}_{=\left( -1\right) ^{n\left( n-1\right) /2} V_{n}}\\ & =s_{\mu}V_{n}\cdot\left( -1\right) ^{n\left( n-1\right) /2}V_{n}=\left( -1\right) ^{n\left( n-1\right) /2}s_{\mu}\cdot V_{n}^{2}. \end{align*} In view of \eqref{darij1.eq.AB=P}, we can rewrite this as \begin{align*} \det P=\left( -1\right) ^{n\left( n-1\right) /2}s_{\mu}\cdot V_{n}^{2}. \end{align*} This proves Theorem 1. $$\blacksquare$$

The claim of Theorem 1 can further be rewritten by observing that (in $$n$$ indeterminates $$x_{1},x_{2},\ldots,x_{n}$$) we have \begin{align*} s_{\lambda}=s_{\mu}\cdot\left( x_{1}x_{2}\cdots x_{n}\right) ^{n-1} \end{align*} (because the entries of $$\lambda$$ are the respective entries of $$\mu$$ plus $$n-1$$). The product $$x_{1}x_{2}\cdots x_{n}$$ can also be rewritten as $$s_{\left( 1^{n}\right) }$$, where $$\left( 1^{n}\right)$$ is the partition $$\left( 1,1,\ldots,1\right)$$ with $$n$$ entries.

• Thank you. The (2) is very nice
– Leox
Jan 21, 2022 at 18:20
• Is there a skew version of this? Mar 18, 2022 at 19:03