$(q,t)$-chromatic symmetric functions
Tatsuyuki Hikita
TL;DR
This work introduces a two-parameter ($q,t$) extension of chromatic symmetric functions for unit interval graphs via level-one affine Hecke algebra representations. It constructs ${f X}_{oldsymbol Gamma}(q,t)$ and a quantum multiplication $\ imes$-deformation from a center isomorphism ${oldsymbol{ m q}}$, linking ${f X}_{oldsymbol Gamma}(q,t)$ to the chromatic quasisymmetric function ${f Y}_{oldsymbol Gamma}(t)$ through ${oldsymbol{ m q}}({f Y}_{oldsymbol Gamma}(t))$, and ensuring symmetry and multiplicativity under the new product. Key results include a Pieri-type rule for $e_1(X)\star e_r(X)$, stabilization to infinite-variable symmetric functions, and a modular-law framework that yields linear relations among $e$-expansion coefficients in the $q=1$ and $q= o\infty$ limits. The $q\to\infty$ limit produces explicit asymptotics ${f X}_{oldsymbol Gamma}(q,t)\to t^{n(n-1)/2-|{oldsymbol e}|}[n]_t! e_n(X)$ and relates to probability interpretations of expansion coefficients, offering a new algebraic pathway to explore the Stanley–Stembridge conjecture and related combinatorics in unit interval graphs.
Abstract
By using level one polynomial representations of affine Hecke algebras of type $A$, we obtain a $(q,t)$-analogue of the chromatic symmetric functions of unit interval graphs which generalizes Syu Kato's formula for the chromatic symmetric functions of unit interval graphs. We show that at $q=1$, the $(q,t)$-chromatic symmetric functions essentially reduce to the chromatic quasisymmetric functions defined by Shareshian-Wachs, which in particular gives an algebraic proof of Kato's formula. We also give an explicit formula of the $(q,t)$-chromatic symmetric functions at $q=\infty$, which leads to a probability theoretic interpretation of $e$-expansion coefficients of chromatic quasisymmetric functions used in our proof of the Stanley-Stembridge conjecture. Moreover, we observe that the $(q,t)$-chromatic symmetric functions are multiplicative with respect to certain deformed multiplication on the ring of symmetric functions. We give a simple description of such multiplication in terms of the affine Hecke algebras of type $A$. We also obtain a recipe to produce $(q,t)$-chromatic symmetric functions from chromatic quasisymmetric functions, which actually makes sense for any oriented graphs.