Related papers: Macdonald Identities: revisited
We use derivatives to prove the equivalences between MacWilliams identity and its four equivalent forms, and present new interpretations for the four equivalent forms. Our results explicitly give out the relationships between MacWilliams…
In this paper we prove the identity that generalizes the Andrews-Gordon identity. Also we discuss the relation of our formula to the geometry of affine flag varieties and to the geometry of polyhedra.
We give the explicit analytic development of Macdonald polynomials in terms of "modified complete" and elementary symmetric functions. These expansions are obtained by inverting the Pieri formula. Specialization yields similar developments…
We propose and develop a theory that allows to characterize epimorphisms of profinite groups in terms of indecomposable epimorphisms.
The symmetric Macdonald polynomials are able to be constructed out of the non-symmetric Macdonald polynomials. This allows us to develop the theory of the symmetric Macdonald polynomials by first developing the theory of their non-symmetric…
We prove identities on compound matrices in extended tropical semirings. Such identities include analogues to properties of conjugate matrices, powers of matrices and~$\adj(A)\det(A)^{ -1}$, all of which have implications on the eigenvalues…
We lift to the multivariate Eulerian polynomials the identity implying that univariate Eulerian polynomials are palindromic. As a consequence of this generalization, we obtain nice combinatorial identities that can be directly extracted…
We introduce a family of rank-one local systems in the category of twisted $\mathcal{D}$-modules on a certain subvariety isomorphic to ${\mathbb{G}_{\text{m}}}^2$ of the affine flag variety of $\text{SL}_2$. We then give a criterion for…
We prove analogues for elliptic interpolation functions of Macdonald's version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic "hypergeometric" sums as skew…
We define the analogue of Jack's (Jacobi) polynomials, which were defined for finite-dimensional root system by Heckman and Opdam as eigenfunctions of trigonometric Sutherland operator for the affine root system $\hat A_{n-1}$. In the…
The Capelli identities claim $det(A)det(B) = det(AB+correction)$ for certain matrices with noncommutative entries. They have applications in representation theory and integrable systems. We propose new examples of these identities,…
We consider discrete-time switching systems composed of a finite family of affine sub-dynamics. First, we recall existing results and present further analysis on the stability problem, the existence and characterization of compact…
We give an explicit formula for an operator that sends a wreath Macdonald polynomial to the delta function at a character associated to its partition. This allows us to prove many new results for wreath Macdonald polynomials, especially…
In this note, we will give a short proof of an identity for cubic partitions.
Given an odd prime p, we present three independent ways of relating modulo p certain truncated convolutions of divided Bernoulli numbers to certain full convolutions of divided Bernoulli numbers.
We will prove an identity involving refined $q$-trinomial coefficients. We then extend this identity to two infinite families of doubly bounded polynomial identities using transformation properties of the refined $q$-trinomials in an…
Identities between Whittaker and modified Bessel functions are derived for particular complex orders. Certain polynomials appear in such identities, which satisfy a fourth order differential equation (not of hypergeometric type), and they…
We introduce Macdonald characters and use algebraic properties of Macdonald polynomials to study them. As a result, we produce several formulas for Macdonald characters, which are generalizations of those obtained by Gorin and Panova in…
Using matrix function theory, Perron-Frobenius theory, combinatorial matrix theory, and elementary number theory, we characterize, classify, and describe in terms of the Jordan canonical form the matrix pth-roots of imprimitive irreducible…
In this note, we establish the periodicity of chains of affine standard Lyndon words in all types and determine tight bounds on that periodicity, greatly generalizing the $A$-type results of arXiv:2305.16299. Our approach crucially utilizes…