리대수 지표의 행렬식 표현

개요

• 단순리대수 $B_n$과 $C_n$의 지표는 다음과 같은 행렬식으로 주어짐

$$ \begin{align} \operatorname{so}_{2n+1,\lambda}(x)&=\frac{\det_{1\leq i,j\leq n} \big(x_i^{j-1-\lambda_j}-x_i^{2n-j+\lambda_j}\big)} {\Delta_{\mathrm{B}}(x)}, \\ \operatorname{symp}_{2n,\lambda}(x)&=\frac{\det_{1\leq i,j\leq n} \big(x_i^{j-1-\lambda_j}-x_i^{2n-j+1+\lambda_j}\big)} {\Delta_{\mathrm{C}}(x)}. \end{align} $$ 여기서 $\Delta_{\mathrm{B}}$ 와 $\Delta_{\mathrm{C}}$ 일반화된 반데몬드 행렬식 $$ \begin{align*} \Delta_{\mathrm{B}}(x)&:=\prod_{i=1}^n (1-x_i) \prod_{1\leq i<j\leq n} (x_i-x_j)(x_ix_j-1) \\ \Delta_{\mathrm{C}}(x)&:=\prod_{i=1}^n (1-x_i^2) \prod_{1\leq i<j\leq n} (x_i-x_j)(x_ix_j-1). \end{align*} $$

메모

• Krattenthaler, C. “Identities for Classical Group Characters of Nearly Rectangular Shape.” Journal of Algebra 209, no. 1 (1998): 1–64. doi:10.1006/jabr.1998.7531.
• Okada, Soichi. “Applications of Minor Summation Formulas to Rectangular-Shaped Representations of Classical Groups.” Journal of Algebra 205, no. 2 (1998): 337–67. doi:10.1006/jabr.1997.7408.