Related papers: From Cauchy's determinant formula to bosonic and f…

Some identities that involve the elliptic version of the Cauchy matrices are presented and proved. They include the determinant formula, the formula for the inverse matrix, the matrix product identity and the factorization formula.

In this note, we present a simple proof of an analogue of the Cauchy-Schwarz inequality relevant to products of determinants. Specifically, we show that $$ |\det(A^*MB)|^2\leq \det(A^*MA)\cdot \det(B^*MB),\quad A,B\in \mathbb{C}^{m\times…

We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor…

In this paper we study the integrability of a family of models with U(1)xSU(N) symmetry. They admit fermionic and bosonic formulations related through bosonization and subsequent T-duality. The fermionic theory is just the CP^(N-1) sigma…

Representing fermionic wavefunctions efficiently is a central problem in quantum physics, chemistry and materials science. In this work, we introduce a universal and exact representation of continuous antisymmetric functions by lifting them…

The classical Menon's identity [7] states that \begin{equation*}\label{oldbegin1} \sum_{\substack{a\in\Bbb Z_n^\ast }}\gcd(a -1,n)=\varphi(n) \sigma_{0} (n), \end{equation*} where for a positive integer $n$, $\Bbb Z_n^\ast$ is the group of…

We prove an interesting identity for the sum of determinants, which is a generalization of the sum of a geometric progression. The proof is quite long and a number of other identities are proved along the way. Some of the more elementary…

We fully describe the general form of a linear (or conjugate-linear) rank metric isometry on the Murray--von Neumann algebra associated with a II$_1$-factor. As an application, we establish Frobenius' theorem in the setting of…

The foundation for the theory of correlation functions of exactly solvable models is determinant representation. Determinant representation permit to describe correlation functions by classical completely integrable differential equations…

This paper explores a quantum deformation of the classical identity Pf(A)^2 = det(A) for 2n by 2n skew-symmetric matrices A, which classically relates the square of the Pfaffian to the determinant. In the quantum setting, we study matrices…

We prove that, for $X$, $Y$, $A$ and $B$ matrices with entries in a non-commutative ring such that $[X_{ij},Y_{k\ell}]=-A_{i\ell} B_{kj}$, satisfying suitable commutation relations (in particular, $X$ is a Manin matrix), the following…

Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only…

Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms of powers of the…

In this paper, we describe the dynamical symmetries of classical supersymmetric oscillators in one and two spatial (bosonic) dimensions. Our main ingredient is a generalized Poisson bracket which is defined as a suitable classical…

We investigate the statistical properties of $C=uvu^{-1}v^{-1}$, when $u$ and $v$ are independent random matrices, uniformly distributed with respect to the Haar measure of the groups $U(N)$ and $O(N)$. An exact formula is derived for the…

The polynomial relationship between elementary symmetric functions (Cauchy enumeration formula) is formulated via a ``raising operator" and Fock space construction. A simple graphical proof of this relation is proposed. The new operator…

We consider two sequences $a(n)$ and $b(n)$, $1\leq n<\infty$, generated by Dirichlet series $$\sum_{n=1}^{\infty}\frac{a(n)}{\lambda_n^{s}}\qquad\text{and}\qquad \sum_{n=1}^{\infty}\frac{b(n)}{\mu_n^{s}},$$ satisfying a familiar functional…

We consider a Dirac field in 2+1 Euclidean dimensions, in the presence of a linear domain wall defect in its mass, and a constant electromagnetic field. We evaluate the exact fermionic determinant for the situation where the defect is…

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of…

Let $\mathbb{A}_n^m$ be an arbitrary $n$-dimensional commutative associative algebra over the field of complex numbers with $m$ idempotents. Let $e_1=1,e_2,\ldots,e_k$ with $2\leq k\leq 2n$ be elements of $\mathbb{A}_n^m$ which are linearly…