Notes about KP/BKP correspondence

TL;DR

This paper discusses various mathematical remarks and new results related to KP/BKP correspondence, including polynomial solutions, vertex operators, eigenproblems, and bilinear relations among symmetric functions, with implications for matrix models and conjectures.

Contribution

It introduces new bilinear relations between characters of symmetric and Sergeev groups, and between different types of Schur functions, expanding understanding of KP/BKP correspondence.

Findings

New bilinear relations between symmetric group characters

Relations between skew, shifted, and projective Schur functions

Analysis of matrix models and cut-and-join relations

Abstract

I present a set of remarks related to joint works \cite{paper1},\cite{paper2},\cite{paper3},\cite{MMNO}. These are remarks about polynomials solutions and vertex operators, eigenproblem for polynomials and a remark related to the the conjecture of Alexandrov and Mironov, Morozov about the ratios of the projective Schur functions. New results on the bilinear relations between characters of the symmetric and of the Sergeev group and on bilinear relations between skew Schur and projective Schur functions and also between shifted Schur and projective Schur functions are added. Certain new matrix models are considered and a comment on Mironov-Morozov-Natanzon cut-and-join relation is added.