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The Schur polynomial in $d$ variables $s_\lambda (x_1,x_2,\ldots,x_d)$ for an integer partition $\lambda$ of $k$, has a simple form when we evaluate all arguments at one, i.e. $s_\lambda (1,\ldots,1)$. We can write the expression in terms of the character of the identity as $$ s_\lambda (1,\ldots,1) = \frac{1}{k!} \chi^\lambda(\mathbb{1}) z_\lambda(d)\,, \quad {\rm where} \quad z_\lambda(d) = \prod_{(i,j)\in \lambda} (d+j-i) $$ with the product taken over the Young diagram of $\lambda$. This is often written more explicitly as $$ s_\lambda (1,\ldots,1) = \prod_{1\leq i<j \leq d} \frac{\lambda_i - \lambda_j +j-i}{j-i}\,. $$ For our purposes, given a partition $\lambda$, $s_\lambda (1,\ldots,1)$ is just some polynomial in $d$. For example, the partition $\lambda=\{2,1\}$ gives $s_\lambda(1,\ldots,1) = \frac{1}{3}d (d^2-1)$.

I am interested in Schur polynomials of the form $s_\lambda(1,\ldots,1,-1,\ldots,-1)$, i.e. a polynomial of $d$ arguments, evaluated on $a$ 1's and $b$ -1's, where $a+b=d$. Is there a simple expression for $s_\lambda(1_a,-1_b)$ as a polynomial in $a$ and $b$?

I am aware that we may combine Schur functions using the Littlewood-Richardson rule as $s_\lambda(x,y) = \sum_{\mu,\nu} c^\lambda_{\mu\nu} s_\mu (x) s_\nu (y)$. By computing the coefficients and using the expressions above it seems I can find the expression I want. Is there a simpler way to compute $s_\lambda(1_a,-1_b)$? Also, is there a simple way to compute the coefficients $c^\lambda_{\mu\nu}$ in terms of characters $\chi^\lambda$?

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Posting an answer in case anyone was interested. There doesn't seem to be a nice compact expression as in the all 1's case. But there is a simpler approach that doesn't involve Littlewood-Richardson coefficients, but instead just requires locating the right expression in Macdonald's 'Symmetric Functions and Hall Polynomials' text.

The Schur polynomial may be written as $$ s_\lambda(x_1,\ldots,x_d) = \sum_{\rho \vdash k} z^{-1}_\rho \chi^\lambda_\rho \, p_\rho(x_1,\ldots,x_d) $$ where $\chi^\lambda_\rho$ is the character $\chi^\lambda(\sigma)$ evaluated on elements of $S_k$ of cycle-type $\rho$, $z_\lambda = \prod_i i^{m_i} m_i!$, where $m_i$ is the multiplicity of $i$ in the partition, and $p_\rho$ is the power-sum symmetric polynomial on the partition $\rho$. Knowing this, we can simply write $$ s_\lambda(1,\ldots,1,-1,\ldots,-1) = \sum_{\rho \vdash k} z^{-1}_\rho \chi^\lambda_\rho \, \textstyle\prod_{\rho_i} (a+(-1)^{\rho_i} b)\,, $$ which gives a polynomial in $a$ and $b$, as desired. For example, the partition $\lambda=\{1,1\}$ gives $s_\lambda(1_a,-1_b)=\frac{1}{2}((a - b)^2-(a+b))$.

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