A Spin Analog of the Plethystic Murnaghan–Nakayama Rule

Abstract

As a spin analog of the plethystic Murnaghan–Nakayama rule for Schur functions, the plethystic Murnaghan–Nakayama rule for Schur Q-functions is established with the help of the vertex operator realization. This generalizes both the Murnaghan–Nakayama rule and the Pieri rule for Schur Q-functions. A plethystic Murnaghan–Nakayama rule for Hall–Littlewood functions is also investigated.

Appendix

Proof of (3.14):

$B=(b_{ij})_{2n\times 2n}$ is an antisymmetric matrix obtained from $A=(a_{ij})_{2n\times 2n}$ by rotating the first i rows and columns by the cyclic permutation $\left( \begin{array}{ccccc} 1&{}2&{}\cdots &{}i-1&{}i\\ 2&{}3&{}\cdots &{}i&{}1\\ \end{array}\right) .$ Explicitly

$$\begin{aligned} b_{1j+1}= & {} {\left\{ \begin{array}{ll} a_{ij}&{} \text { if } j=1,2,\ldots ,i-1;\\ a_{i,j+1}&{} \text { if } j=i,i+1,\ldots ,2n. \end{array}\right. }\\ \quad B_{1j+1}= & {} {\left\{ \begin{array}{ll} A_{ij}&{} \text {if } j=1,2,\ldots ,i-1;\\ A_{i,j+1}&{} \text {if } j=i,i+1,\ldots ,2n. \end{array}\right. } \end{aligned}$$

Therefore $\text {Pf(A)}=(-1)^{i-1}\text {Pf(B)}.$ It follows from (3.13) that

$$\begin{aligned} \text {Pf(B)}&=\sum _{j=2}^{2n}(-1)^{j}b_{1j}\text {Pf}(B_{1j})\\&=\sum _{j=1}^{i-1}(-1)^{j+1}b_{1j+1}\text {Pf}(B_{1j+1})+\sum _{j=i+1}^{2n}(-1)^{j}b_{1j}\text {Pf}(B_{1j})\\&=\sum _{j=1}^{i-1}(-1)^{j+1}a_{ij}\text {Pf}(A_{ij})+\sum _{j=i+1}^{2n}(-1)^{j}a_{ij}\text {Pf}(A_{ij})\\&=\sum _{j=1}^{i-1}(-1)^{j}a_{ji}\text {Pf}(A_{ij})+\sum _{j=i+1}^{2n}(-1)^{j}a_{ij}\text {Pf}( A_{ij})\\&=\sum \limits _{j\ne i}(-1)^{j}:a_{ij}:\text {Pf}(A_{ij}). \end{aligned}$$

$\square$