Symmetric discrete AKP and BKP equations

结合代数和可积系统的离散q-差变形

Discrete, q-difference deformations of associative algebras and integrable systems

研究了一类结合非交换代数结构常数的离散变形和q-差变形。它示出，这些变形是由一个中央系统的离散或q-差分方程，在特定情况下，代表离散和q-差分版本的定向结合方程。它还表明，著名的Hirota-Miwa双线性方程的AKP和BKP的层次描述某些有限维代数的离散变形。

On reductions of the discrete Kadomtsev--Petviashvili-type equations

The reduction by restricting the spectral parameters $k$ and $k'$ on a generic algebraic curve of degree $\mathcal{N}$ is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable systems possessing a more general solution structure arise from the reduction, and in each case a unified formula for generic positive integer $\mathcal{N}\geq 2$ is given to express the corresponding reduced integrable lattice equations. The obtained extended two-dimensional lattice models give rise to many important integrable partial difference equations …

From the $B$-Toda to the BKP hierarchy

It is shown that all $τ$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $τ$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM is a disjoint union of Schubert cells, we classify and give explicit parameterization for points in each Schubert cell by constructing a frame for UOGM in the sense of Sato. $τ$-functions are then expressed in terms of these frames and Schur-Q functions. For concreteness we give a comprehensive study for the $τ$-functions of $B$-Toda which can be viewed as a finit…

Bilinear expansions of lattices of KP $τ$-functions in BKP $τ$-functions: a fermionic approach

We derive a bilinear expansion expressing elements of a lattice of KP $τ$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $τ$-functions, labelled by strict partitions. This generalizes earlier results relating determinants and Pfaffians of minors of skew symmetric matrices, with applications to Schur functions and Schur $Q$-functions. It is deduced using the representations of KP and BKP $τ$-functions as vacuum expectation values (VEV's) of products of fermionic operators of charged and neutral type, respectively. The lattice is ge…

关于 c-k 约束 KP 与 BKP 层级：费米子图像、解与附加对称

On the c-k constrained KP and BKP hierarchies: the Fermionic pictures, solutions and additional symmetries

本文通过费米子方法研究 c-k 约束 KP 与 BKP 层级，给出费米子图像、解及附加对称，并证明其附加流均构成 Virasoro 子代数。

Notes about KP/BKP correspondence

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 m…

$b$-单调Hurwitz数：Virasoro约束、BKP可积层级与$O(N)$-BGW积分

$b$-monotone Hurwitz numbers: Virasoro constraints, BKP hierarchy, and $O(N)$-BGW integral

本文研究单调Hurwitz数的$b$-形变，推导其Virasoro约束并证明相关猜想，在$b=1$时得到Schur展开并验证BKP可积层级，还将模型与$O(N)$版BGW积分关联并显式求解。

与局部Yang-Baxter关系相联系的三维离散方程

On Discrete 3-Dimensional Equations Associated with the Local Yang-Baxter Relation

Maillet和Nijhoff提出的局部杨-巴克斯特方程（YBE）是零曲率关系在三维空间的适当推广。最近，Korepanov构造了一组无限的可积三维晶格模型，并将它们与局部YBE的解联系起来。最简单的Korepanov模型与伊辛模型中的星三角关系有关。本文导出了相应的离散方程。在连续极限中，它导致一个微分3d方程，该方程关于三个坐标的所有排列对称。对电网络中的星三角变换的类似分析导致与BKP族相关联的离散双线性Miwa方程。并构造了四面体方程的相关算子解。