리대수 지표의 행렬식 표현

개요

• 자연수 $n$을 고정
• 분할 \(\lambda: \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\geq 0\)
• 슈르 다항식(Schur polynomial)은 다음과 같이 행렬식을 이용하여 정의된다

\[s_{\lambda} = \frac{\det_{1\le i,j\le n}(x_{i}^{\lambda_{j}+n-j})}{\Delta(x)} \label{van}\]

• 여기서 분모는 반데몬드 행렬과 행렬식 (Vandermonde matrix)에서 등장하는 반데몬드 다항식이다

\[\Delta(x):=\det_{1\le i,j\le n}(x_{i}^{n-j})=\prod_{1\le i<j\le n} (x_i-x_j)\]

• 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 $$ \begin{align*} \Delta_{\mathrm{B}}(x)&:=\prod_{i=1}^n (1-x_i) \prod_{1\leq i<j\leq n} (x_i-x_j)(x_ix_j-1) \\ \Delta_{\mathrm{C}}(x)&:=\prod_{i=1}^n (1-x_i^2) \prod_{1\leq i<j\leq n} (x_i-x_j)(x_ix_j-1). \end{align*} $$

리틀우드 항등식

• $m,n\in \mathbb{Z}_{\geq 0}$, $n\geq 1$
• typc B (Macdonald)

$$ (x_1\cdots x_n)^{m/2} \operatorname{so}_{2n+1,(\frac{m}{2})^n}(x)=\sum_{\substack{\lambda \\[1.5pt] \lambda_1\leq m}} s_{\lambda}(x) $$

• type C (Désarménien-Proctor-Stembridge)

$$ (x_1\cdots x_n)^m \operatorname{symp}_{2n,(m^n)}(x)=\sum_{\substack{\lambda \text{ even} \\[1.5pt] \lambda_1\leq 2m}} s_{\lambda}(x) $$

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. http://arxiv.org/abs/math/9808118
• 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.

관련된 항목들

• 행렬식

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxbFVlaTZNbUN6cDQ/edit

관련도서

• “Symmetric Functions and Hall Polynomials (Oxford Mathematical Monographs): I. G. Macdonald: 9780198504504: Amazon.com: Books.” http://www.amazon.com/Symmetric-Functions-Polynomials-Mathematical-Monographs/dp/0198504500.

관련논문

• Fulmek, Markus, and Christian Krattenthaler. “Lattice Path Proofs for Determinantal Formulas for Symplectic and Orthogonal Characters.” Journal of Combinatorial Theory. Series A 77, no. 1 (1997): 3–50. doi:10.1006/jcta.1996.2711.
• Hamel, A. M. “Determinantal Forms for Symplectic and Orthogonal Schur Functions.” Canadian Journal of Mathematics. Journal Canadien de Mathématiques 49, no. 2 (1997): 263–82. doi:10.4153/CJM-1997-013-5.