Related papers: Demazure crystal structure for flagged reverse pla…

Catalan functions, the graded Euler characteristics of certain vector bundles on the flag variety, are a rich class of symmetric functions which include $k$-Schur functions and parabolic Hall-Littlewood polynomials. We prove that Catalan…

We describe two crystal structures on set-valued decomposition tableaux. These provide the first examples of interesting "$K$-theoretic" crystals on shifted tableaux. Our first crystal is modeled on a similar construction of Monical,…

For a classical simple algebraic group $G$ we obtain the affirmative answer for the conjecture in [8] that there exists an isomorphism between the geometric crystal on the flag variety and the one on the unipotent subgroup $U^-$.

This paper provides a combinatorial characterisation for generic forced symmetric rigidity of bar-joint frameworks in the Euclidean plane that are symmetric with respect to the orientation-reversing wallpaper group…

This paper is the sequel of the paper under the same title with part 1, where we introduced refined canonical stable Grothendieck polynomials and their duals with two families of infinite parameters. In this paper we give combinatorial…

We give positive descriptions for certain Schubert structure constants $c_{u,v}^w$ for the full flag variety in Lie types $C$ and $D$. This is accomplished by first observing that a number of the $K=GL(n,\C)$-orbit closures on these flag…

Given a weighted graph $G$ embedded in a non-orientable surface $\Sigma$, one can consider the corresponding weighted graph $\widetilde{G}$ embedded in the so-called orientation cover $\widetilde\Sigma$ of $\Sigma$. We prove identities…

A purely combinatorial construction of the quantum cohomology ring of the flag manifold $G/B$ is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of two of…

We show that the totally nonnegative part of the twisted product of flag varieties of a Kac-Moody group admits a cellular decomposition, and the closure of each cell is a topological manifold with boundary. We also establish explicit…

We shall realize certain affine geometric crystal of type $D_4^{(3)}$ associated with the fundamental representation $W(\pi_1)$ explicitly . By its explicit form, we see that it has a positive structure.

Fix a scheme $S$ of characteristic $p$. Let $\mathscr{M}$ be an $S$-algebraic stack and let $\mbox{Fdiv}(\mathscr{M})$ be the stack of $\mbox{F}$-divided objects, that is sequences of objects $x_i\in\mathscr{M}$ with isomorphisms…

Let $\mathcal{X}$ be a tame proper Deligne-Mumford stack of the form $[M/G]$ where $M$ is a scheme and $G$ is an algebraic group. We prove that the stack $\mathcal{K}_{g,n}(\mathcal{X},d)$ of twisted stable maps is a quotient stack and can…

We study the back stable $K$-theory Schubert calculus of the infinite flag variety. We define back stable (double) Grothendieck polynomials and double $K$-Stanley functions and establish coproduct expansion formulae. Applying work of…

We give an alternative proof of a (former) conjecture of Bj\"orner stating that the matrix expressing face numbers in terms of g numbers is totally non-negative. We briefly discuss the case of simple flag polytopes.

We apply crystal theory to affine Schubert calculus, Gromov-Witten invariants for the complete flag manifold, and the positroid stratification of the positive Grassmannian. We introduce operators on decompositions of elements in the…

We study closed, oriented 4-manifolds whose fundamental group is that of a closed, oriented, aspherical 3-manifold. We show that two such 4-manifolds are stably diffeomorphic if and only if they have the same w_2-type and their equivariant…

We classify homogeneous reversible Finsler metrics with positive Flag curvature. We show that if G/H admits a G invariant reversible Finsler metric with positive Flag curvature, then up to a few low dimensional spaces, it also admits a G…

We consider the PBW filtrations over the integers of the irreducible highest weight modules in type A and C. We show that the associated graded modules can be realized as Demazure modules for group schemes of the same type and doubled rank.…

The flag curvature is a natural Finsler extension of the sectional curvature in Riemannian geometry. However, there are many non-Riemannian quantities which interact with the flag curvature. In this paper, we introduce a notion of weighted…

We define a family of symmetric polynomials $G_{\nu,\lambda}(z_1,\cdots, z_{n+1},q)$ indexed by a pair of dominant integral weights. The polynomial $G_{\nu,0}(z,q)$ is the specialized Macdonald polynomial and we prove that…