Related papers: 2D Toda \tau-functions as combinatorial generating…
The subject of this paper is a connection between d-orthogonal polynomials and the Toda lattice hierarchy. In more details we consider some polynomial systems similar to Hermite polynomials, but satisfying $d+2$-term recurrence relation, $d…
We derive two multivariate generating functions for three-dimensional Young diagrams (also called plane partitions). The variables correspond to a colouring of the boxes according to a finite Abelian subgroup G of SO(3). We use the vertex…
A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate…
A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional ``phase space'' variables $(k,x)$ of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions…
We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational…
We compute the ${\cal N}=2$ supersymmetric partition function of a gauge theory on a four-dimensional compact toric manifold via equivariant localization. The result is given by a piecewise constant function of the K\"ahler form with jumps…
A continuum limit of the Toda lattice field theory, called the SDiff(2) Toda equation, is shown to have a Lax formalism and an infinite hierarchy of higher flows. The Lax formalism is very similar to the case of the self-dual vacuum…
By exhibiting the corresponding Lax pair representations we propose a wide class of integrable two-dimensional (2D) fermionic Toda lattice (TL) hierarchies which includes the 2D N=(2|2) and N=(0|2) supersymmetric TL hierarchies as…
We show that the 4-variable generating function of certain orientation related parameters of an ordered oriented matroid is the evaluation at (x + u, y+v) of its Tutte polynomial. This evaluation contains as special cases the counting of…
Plethysm coefficients $\mathsf{a}_{\mu[\nu]}^\lambda$ are the structure coefficients of the plethysm of Schur functions $s_\mu[s_\nu] = \sum_{\lambda} \mathsf{a}_{\mu[\nu]}^\lambda s_\lambda$. We study a bivariate generating function of…
Liedtke (2008) has introduced group functors $K$ and $\tilde K$, which are used in the context of describing certain invariants for complex algebraic surfaces. He proved that these functors are connected to the theory of central extensions…
We study the WKB expansion of $2\times 2$ system of linear differential equations with four fuchsian singularities. The main focus is on the generating function of the monodromy symplectomorphism which, according to a recent paper is…
We examine the coefficients in front of Chern numbers for complex genera, and pay special attention to the $\text{Td}^{\frac{1}{2}}$-genus, the $\Gamma$-genus as well as the Todd genus. Some related geometric applications to…
We derive a bilinear expansion expressing elements of a lattice of KP $\tau$-functions, labelled by partitions, as a sum over products of pairs of elements of an associated lattice of BKP $\tau$-functions, labelled by strict partitions.…
We present a family of matrix models such that their partition functions are tau functions of the universal character (UC) hierarchy. This develops one of the topics of our previous paper arXiv:2410.14823. We found new matrix models…
The $K$-theoretic Schur $P$- and $Q$-functions $GP_\lambda$ and $GQ_\lambda$ may be concretely defined as weight generating functions for semistandard shifted set-valued tableaux. These symmetric functions are the shifted analogues of…
We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric…
The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are…
We study Hilbert schemes of points on a smooth projective Calabi-Yau 4-fold $X$. We define $\mathrm{DT}_4$ invariants by integrating the Euler class of a tautological vector bundle $L^{[n]}$ against the virtual class. We conjecture a…
The enumeration of planar maps equipped with an Eulerian orientation has attracted attention in both combinatorics and theoretical physics since at least 2000. The case of 4-valent maps is particularly interesting: these orientations are in…