Related papers: On Factorization of Generalized Macdonald Polynomi…

The Douglas' majorization and factorization theorem characterizes the inclusion of operator ranges in Hilbert spaces. Notably, it reinforces the well-established connections between the inclusion of kernels of operators in Hilbert spaces…

Quantum analogues of the homogeneous spaces $\GL(n)/\SO(n)$ and $\GL(2n)/\Sp(2n)$ are introduced. The zonal spherical functions on these quantum homogeneous spaces are represented by Macdonald's symmetric polynomials…

Factorization theorems for single inclusive jet production play a crucial role in the study of jets and their substructure. In the case of small radius jets, the dynamics of the jet clustering can be factorized from both the hard production…

We introduce a class of Schur type functions associated with polynomial sequences of binomial type. This can be regarded as a generalization of the ordinary Schur functions and the factorial Schur functions. This generalization satisfies…

In this article we consider means of positive operators on a Hilbert space. We extend the theory of matrix power means to arbitrary operator means in the sense of Kubo-Ando. The basis of the extension is relying on ideas coming from…

$Q$-systems and $T$-systems are systems of integrable difference equations that have recently attracted much attention, and have wide applications in representation theory and statistical mechanics. We show that certain $\tau$-functions,…

We describe majorization between selfadjoint operators in a $\sigma$-finite II$_\infty$ factor $(\mathcal{M},\tau)$ in terms of simple spectral relations. For a diffuse abelian von Neumann subalgebra $\mathcal{A}\subset \mathcal{M}$ with…

The Schur class, denoted by $\mathcal{S}(\mathbb{D})$, is the set of all functions analytic and bounded by one in modulus in the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$, that is \[ \mathcal{S}(\mathbb{D}) = \{\varphi…

We prove that the Buchweitz-Greuel-Schreyer Conjecture on the minimal rank of a matrix factorization holds for a generic polynomial of given degree and strength. The proof introduces a notion of the secondary strength of a polynomial, and…

We argue that Jack Littlewood-Richardson coefficients $g_{\mu\nu}^{\lambda}(\alpha)$ are specialisations of certain novel polynomials. For the triple of partitions $(\mu,\nu,\lambda)=(21,21,321)$, we prove the corresponding polynomial is…

Schur functions provide an integral basis of the ring of symmetric functions. It is shown that this ring has a natural Hopf algebra structure by identifying the appropriate product, coproduct, unit, counit and antipode, and their…

We prove the equivalence of a class of generalised Schur partition functions $\mathcal Z_G(q;\alpha)$ of 4d $\mathcal N=2$ superconformal gauge theories to contour integral representations of vector-valued modular forms of the type that…

Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance: $ \bullet $ its value in $1$…

We consider the deformations of ``monomial solutions'' to Generalized Kontsevich Model \cite{KMMMZ91a,KMMMZ91b} and establish the relation between the flows generated by these deformations with those of $N=2$ Landau-Ginzburg topological…

We introduce two families of symmetric functions generalizing the factorial Schur $P$- and $Q$- functions due to Ivanov. We call them $K$-theoretic analogues of factorial Schur $P$- and $Q$- functions. We prove various combinatorial…

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on…

It is an important problem in algebraic combinatorics to deduce the Schur function expansion of a symmetric function whose expansion in terms of the fundamental quasisymmetric function is known. For example, formulas are known for the…

Generalized convolution symmetries of integrable hierarchies of KP and 2KP-Toda type multiply the Fourier coefficients of the elements of the Hilbert space $\HH= L^2(S^1)$ by a specified sequence of constants. This induces a corresponding…

A classical result of Littlewood gives a factorisation for the Schur function at a set of variables "twisted" by a primitive $t$-th root of unity, characterised by the core and quotient of the indexing partition. While somewhat neglected,…

The ring of symmetric functions occupies a central place in algebraic combinatorics, with a particularly notable role in Schubert calculus, where the standard cell decompositions of Grassmannians yield the celebrated family of Schur…