arXiv subjects

Severyn Khomych

Publications and source records attributed to Severyn Khomych.

3 recordsLinked to original sources

arXiv2025

Standard Lyndon loop words: weighted orders

We generalize the study of standard Lyndon loop words from [A.Negut, A.Tsymbaliuk, "Quantum loop groups and shuffle algebras via Lyndon words", Adv. Math. 439 (2024), Paper No. 109482] to a more general class of orders on the underlying alphabet, as suggested in Remark 3.15 of loc.cit. The main new ingredient is the exponent-tightness of these words, which also allows to generalize the construction of PBW bases of the untwisted quantum loop algebra via the combinatorics of loop words.

math.RT↗

arXiv2024

The diagonal derivative of a skew Schur polynomial

We prove a formula for the image of a skew Schur polynomial $s_{λ/μ}\left( x_{1}, x_{2}, \ldots, x_{N}\right) $ under the differential operator $\nabla:= \dfrac{\partial}{\partial x_{1}} +\dfrac{\partial}{\partial x_{2}}+\cdots+\dfrac{\partial}{\partial x_{N}}$. This generalizes a formula of Weigandt for $\nabla\left( s_λ\right) $.

math.CO↗

arXiv2023

The Pak--Postnikov and Naruse skew hook length formulas: a new proof

The classical hook length formula of enumerative combinatorics expresses the number of standard Young tableaux of a given partition shape as a single fraction. In recent years, two generalizations of this formula have emerged: one by Pak and Postnikov, replacing the number by a (rational) generating function, and one by Naruse, which generalizes the setting from a partition to a skew partition. Both generalizations appear to lie significantly deeper, with no simple proofs known. We combine them into a generating-function identity for skew partitions, and prove it in a fairly elementary way using recursion, determinants and simple combinatorics.

math.CO↗