리대수 지표의 행렬식 표현

수학노트
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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

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

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


일반화

$B_n$

  • \(\lambda: \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\geq 0\), $\lambda_i$는 모두 정수 또는 정수+1/2 꼴

$$ \operatorname{so}_{2n+1,\lambda}(x)=\frac{\det_{1\leq i,j\leq n} \big(x_i^{\lambda_j+n-j+1/2}-x_i^{-(\lambda_j+n-j+1/2)}\big)}{\det_{1\leq i,j\leq n} \big(x_i^{n-j+1/2}-x_i^{-(n-j+1/2)}\big)}=\frac{\det_{1\leq i,j\leq n} \big(x_i^{j-1-\lambda_j}-x_i^{2n-j+\lambda_j}\big)} {\Delta_{\mathrm{B}}(x)} $$ 여기서 $$ \Delta_{\mathrm{B}}(x)=\prod_{i=1}^n (1-x_i) \prod_{1\leq i<j\leq n} (x_i-x_j)(x_ix_j-1) $$


$C_n$

  • \(\lambda: \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n\geq 0\), $\lambda_i$는 모두 정수

$$ \operatorname{symp}_{2n,\lambda}(x)=\frac{\det_{1\leq i,j\leq n} \big(x_i^{\lambda_j+n-j+1}-x_i^{-(\lambda_j+n-j+1)}\big)}{\det_{1\leq i,j\leq n} \big(x_i^{n-j+1}-x_i^{-(n-j+1)}\big)} =\frac{\det_{1\leq i,j\leq n} \big(x_i^{j-1-\lambda_j}-x_i^{2n-j+1+\lambda_j}\big)}{\det_{1\leq i,j\leq n} \big(x_i^{j-1-\lambda_j}-x_i^{2n-j+1+\lambda_j}\big)} $$ 여기서 $$ \Delta_{\mathrm{C}}(x)=\prod_{i=1}^n (1-x_i^2) \prod_{1\leq i<j\leq n} (x_i-x_j)(x_ix_j-1). $$


$D_n$

  • \(\lambda: \lambda_1 \geq \lambda_2 \geq \cdots \geq |\lambda_n|\geq 0\), $\lambda_i$는 모두 정수 또는 정수+1/2 꼴

$$ \operatorname{so}_{2n,\lambda}(x)=\frac{\det_{1\leq i,j\leq n} \big(x_i^{\lambda_j+n-j}+x_i^{-(\lambda_j+n-j)}\big)+\det_{1\leq i,j\leq n} \big(x_i^{\lambda_j+n-j}-x_i^{-(\lambda_j+n-j)}\big)}{\det_{1\leq i,j\leq n} \big(x_i^{n-j}+x_i^{-(n-j)}\big)} $$

분할과 무게

  • 각 경우의 분할은 다음과 같은 방식으로 리대수 기약표현의 무게에 대응된다

$$ \begin{align} &(\lambda_1-\lambda_2)\omega_1+\cdots+(\lambda_{n-1}-\lambda_n)\omega_{n-1}+2\lambda_n\omega_n, & B_n, \\ &(\lambda_1-\lambda_2)\omega_1+\cdots+(\lambda_{n-1}-\lambda_n)\omega_{n-1}+\lambda_n\omega_n, & C_n, \\ &(\lambda_1-\lambda_2)\omega_1+\cdots+(\lambda_{n-1}-\lambda_n)\omega_{n-1}+ (\lambda_{n-1}+\lambda_n)\omega_n, & D_n, \end{align} $$

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


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관련도서


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.