arXiv subjects
Publications and source records attributed to Tatsuyuki Hikita.
We give a probabilistic interpretation of the coefficients of the elementary symmetric function expansion of the chromatic quasisymmetric function for any unit interval graph. As a corollary, we prove the Stanley--Stembridge conjecture.
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.
We propose several properties of elliptic lifts of the $K$-theoretic canonical bases for conical symplectic resolutions defined in our previous work. As an example, we construct elliptic lifts of canonical bases for the Hilbert scheme of 2-points in the affine plane.
Lusztig defined certain involutions on the equivariant K-theory of Slodowy varieties and gave a characterization of certain bases called canonical bases. In this paper, we give a conjectural generalization of these involutions and K-theoretic canonical bases to conical symplectic resolutions which have good Hamiltonian torus actions and state several conjectures related to them which we check for toric hyper-Kahler manifolds. We also propose an elliptic analogue of these bar involutions. As a verification of our proposal, we explicitly construct elliptic lifts of K-theoretic canonical bases and prove that they are invariant under elliptic bar involutions for toric hyper-Kahler manifolds.
We show that the cohomology ring of Hilbert scheme of $n$-points in the affine plane is isomorphic to the coordinate ring of $\mathbb{G}_{m}$-fixed point scheme of the $n$-th symmetric product of $\mathbb{C}^{2}$ for a natural $\mathbb{G}_{m}$-action on it. This result can be seen as an analogue of a theorem of DeConcini, Procesi and Tanisaki on a description of the cohomology ring of Springer fiber of type A.
We calculate the Borel-Moore homology of affine Springer fibers of type $A$ associated to some regular semisimple nil elliptic elements. As a result, we obtain bigraded $\mf{S}_{n}$-modules whose bigraded Frobenius series are generalization of the symmetric functions introduced by Haglund, Haiman, Loehr, Remmel, and Ulyanov.