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We present combinatorial operators for the expansion of the Kronecker product of irreducible representations of the symmetric group. These combinatorial operators are defined in the ring of symmetric functions and act on the Schur functions…
We investigate Wiener-transformable markets, where the driving process is given by an adapted transformation of a Wiener process. This includes processes with long memory, like fractional Brownian motion and related processes, and, in…
Energy level statistics of Hermitian random matrices $\hat H$ with Gaussian independent random entries $H_{i\geq j}$ is studied for a generic ensemble of almost diagonal random matrices with $ <|H_{ii}|^{2} > \sim 1$ and $<|H_{i\neq j}|^{2}…
It is shown that all $\tau$-functions of BKP hierarchy can be written as Pfaffians of skew-symmetric matrices. $\tau$-functions of BKP hierarchy are parameterized by points in the universal orthogonal Grassmannian manifold (UOGM). The UOGM…
The analogue of Kontsevich's matrix Airy function, with the cubic potential $\operatorname{Tr}\big(\Phi^3\big)$ replaced by a quartic term $\operatorname{Tr}\big(\Phi^4\big)$ with the same covariance, provides a toy model for quantum field…
In this paper we present a closed formula for the CK-extension of the Gaussian distribution in $\mathbb R^m$, and the monogenic version of the holomorphic function $\exp(z^2/2)/z$ which is a fundamental solution of the generalized…
We study the spin Gromov-Witten (GW) theory of $\mathbb{P}^1$. Using the standard torus action on $\mathbb{P}^1$, we prove that the associated equivariant potential can be expressed by means of operator formalism and satisfies the 2-BKP…
Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two…
We define a covariance-type operator on Wiener space: for F and G two random variables in the Gross-Sobolev space $D^{1,2}$ of random variables with a square-integrable Malliavin derivative, we let $Gamma_{F,G}=$ where $D$ is the Malliavin…
Consider the closed convex hull $K$ of a monomial curve given parametrically as $(t^{m_1},\ldots,t^{m_n})$, with the parameter $t$ varying in an interval $I$. We show, using constructive arguments, that $K$ admits a lifted semidefinite…
We extend the theory of distributional kernel operators to a framework of generalized functions, in which they are replaced by integral kernel operators. Moreover, in contrast to the distributional case, we show that these generalized…
For a tau-function of the KP or BKP hierarchy, we introduce the notion of lifting operator and derive an equation connecting the corresponding fermionic two-point function and fermionic one-point function through the lifting operator. This…
The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such…
A recently formulated conjecture of Gamayun, Iorgov and Lisovyy gives an asymptotic expansion of the Jimbo--Miwa--Ueno isomonodromic $\tau$-function for certain Painlev\'e transcendents. The coefficients in this expansion are given in terms…
Let $K$ be a connected compact semisimple group and $V_\lambda$ be an irreducible unitary representation with highest weight $\lambda$. We study the momentum map $\mu:\mathbb P(V_\lambda)\to\mathfrak k^*$. The intersection $\mu(\mathbb…
We construct the ($\beta$-deformed) partition function hierarchies with $W$-representations. Based on the $W$-representations, we analyze the superintegrability property and derive their character expansions with respect to the Schur…
The two function theories of monogenic and of slice monogenic functions have been extensively studied in the literature and were developed independently; the relations between them, e.g. via Fueter mapping and Radon transform, have been…
We generalise the quantum mechanical toy model for gravitationally induced decoherence presented in Xu, Blencowe (2022) and Domi et al. (2024). In contrast to earlier formulations, in which the Hamiltonian of the system of interest is…
We define enumerative invariants associated to a hybrid Gauged Linear Sigma Model. We prove that in the relevant special cases, these invariants recover both the Gromov-Witten type invariants defined by Chang-Li and Fan-Jarvis-Ruan using…
The question of whether given density operators for subsystems of a multipartite quantum system are compatible to one common total density operator is known as the quantum marginal problem. We briefly review the solution of a subclass of…