Tao Gui

• Chinese Academy of Sciences

For any crystallographic root system, let $W$ be the associated Weyl group, and let $\mathit{WP}$ be the weight polytope (also known as the $W$-permutohedron) associated with an arbitrary strongly dominant weight. The action of $W$ on $\mathit{WP}$ induces an action on the toric variety $X(\mathit{WP})$ associated with the normal fan of $\mathit{WP...

We study the cohomology ring of the Bott--Samelson variety. We compute an explicit presentation of this ring via Soergel's result, which implies that it is a combinatorial invariant. We use the presentation to introduce the Bott--Samelson ring associated with a word in arbitrary Coxeter system by generators and relations. In general, it is a split...

Björner and Ekedahl [Ann. of Math. (2), 170(2): 799-817, 2009] pioneered the study of length-counting sequences associated with parabolic lower Bruhat intervals in crystallographic Coxeter groups. In this paper, we study the asymptotic behavior of these sequences in affine Weyl groups. Let W be an affine Weyl group with corresponding finite Weyl gr...

We introduce the dominant weight polytope P λ for a dominant weight λ corresponding to a root system Φ of any Lie type. We provide an explicit formula for the vertices of P λ in terms of the Cartan matrix of Φ. When λ is strongly dominant, we show that P λ is combinatorially equivalent to a rank(Φ)-dimensional hypercube.

We discuss a log-concavity conjecture on the reduced/stable Kronecker coefficients, which is a certain generalization of Okounkov's conjecture on the log-concavity of the Littlewood-Richardson coefficients and the Schur log-concavity theorem of Lam-Postnikov-Pylyavskyy. We prove the conjecture in a special case. We also consider some implications o...

We give new proof of the (generalized) Jacobi-Trudi identity, which expresses the (skew) Schur polynomial as a determinant of the Jacobi-Trudi matrix whose entries are the complete homogeneous symmetric polynomials. The proof is based on interpreting Kostka numbers as tensor product multiplicities in the BGG category $\mathcal{O}$ of $\mathfrak{s l...

We study the $S_n$-equivariant log-concavity of the cohomology of flag varieties, also known as the coinvariant ring of $S_n$. Using the theory of representation stability, we give computer-assisted proofs of the equivariant log-concavity in low degrees and high degrees and conjecture that it holds for all degrees. Furthermore, we make a stronger u...

We show that the exterior algebra $\Lambda_{R}\left[\alpha_{1}, \cdots, \alpha_{n}\right]$, which is the cohomology of the torus $T=(S^{1})^{n}$, and the polynomial ring $\mathbb{R}\left[t_{1}, \ldots, t_{n}\right]$, which is the cohomology of the classifying space $B (S^{1})^{n}=\left(\mathbb{C} \mathbb{P}^{\infty}\right)^{n}$, are $S_{n}$-equivar...