Odd Dunkl Operators and nilHecke Algebras

Symmetric functions appear in many areas of mathematics and physics, including enumerative combinatorics, the representation theory of symmetric groups, statistical mechanics, and the quantum statistics of ideal gases. In the commutative (or "even") case of these symmetric functions, Kostant and Kumar introduced a nilHecke algebra that categorifies the quantum group $U_q(\mathfrak{sl}_2)$. This categorification helps to better understand Khovanov homology, which has important applications in studying knot polynomials and gauge theory. Recently, Ellis and Khovanov initiated the program of "oddification" as an effort to create a representation theoretic understanding of a new "odd" Khovanov homology, which often yields more powerful results than regular Khovanov homology. In this paper, we contribute towards the project of oddification by studying the odd Dunkl operators of Khongsap and Wang in the setting of the odd nilHecke algebra. Specifically, we show that odd divided difference operators can be used to construct odd Dunkl operators, which we use to give a representation of $\mathfrak{sl}_2$ on the algebra of skew polynomials and evaluate the odd Dunkl Laplacian. We then investigate $q$-analogs of divided difference operators to introduce new algebras that are similar to the even and odd nilHecke algebras and act on $q$-symmetric polynomials. We describe such algebras for all previously unstudied values of $q$. We conclude by generalizing a diagrammatic method and developing the novel method of insertion in order to study $q$-symmetric polynomials from the perspective of bialgebras.