Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 089, 12 pages      arXiv:2106.04773      https://doi.org/10.3842/SIGMA.2021.089

Virasoro Action on the $Q$-Functions

Kazuya Aokage ^a, Eriko Shinkawa ^b and Hiro-Fumi Yamada ^c
a) Department of Mathematics, National Institute of Technology, Ariake College, Fukuoka 836-8585, Japan
^b) Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
^c) Department of Mathematics, Kumamoto University, Kumamoto 860-8555, Japan

Received June 10, 2021, in final form October 05, 2021; Published online October 08, 2021

Abstract
A formula for Schur $Q$-functions is presented which describes the action of the Virasoro operators. For a strict partition, we prove a concise formula for $L_{-k}Q_{\lambda}$, where $L_{-k}$ $(k\geq 1)$ is the Virasoro operator.

Key words: $Q$-functions; Virasoro operators.

pdf (370 kb)   tex (13 kb)  

References

1. Aokage K., Shinkawa E., Yamada H.F., Pfaffian identities and Virasoro operators, Lett. Math. Phys. 110 (2020), 1381-1389.
2. Hoffman P.N., Humphreys J.F., Projective representations of the symmetric groups. $Q$-functions and shifted tableaux, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992.
3. Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943-1001.
4. Liu X., Yang C., $Q$-polynomial expansion for Brezin-Gross-Witten tau-function, arXiv:2104.01357.
5. Wakimoto M., Yamada H.-F., The Fock representations of the Virasoro algebra and the Hirota equations of the modified KP hierarchies, Hiroshima Math. J. 16 (1986), 427-441.
6. Yamada H.-F., Reduced Fock representation of the Virasoro algebra, in Proceedings of the 35th Symposium on Algebraic Combinatorics, 2018, 38-45, available at https://hnozaki.jimdofree.com/proceedings-symp-alg-comb/no-35/.