相关论文: Hook Multiplication in the Quantum K-Theory of Gra…

We construct for each choice of a quiver $Q$, a cohomology theory $A$ and a poset $P$ a "loop Grassmannian" $\mathcal{G}^P(Q,A)$. This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms.…

In an earlier paper, two of the authors defined a $5$-vertex Yang-Baxter algebra (a Hopf algebra) which acts on the sum of the equivariant quantum K-rings of Grassmannians $\mathrm{Gr}(k;n)$, where $k$ varies from $0$ to $n$. We construct…

We prove a twisting theorem for nodal classes in permutation-equivariant quantum $K$-theory, and combine it with existing theorems of Givental to obtain a twisting result for general characteristic classes of the virtual tangent bundle.…

We prove a formula expressing the $K$-theoretic log Gromov-Witten invariants of a product of log smooth varieties $V \times W$ in terms of the invariants of $V$ and $W$. The proof requires introducing log virtual fundamental classes in…

$GQ$ functions are symmetric functions indexed by strict partitions that represent $K$-theoretic Schubert classes in the Lagrangian Grassmannian. Buch and Ravikumar proved a Pieri rule for expanding $GQ_{\lambda}\cdot GQ_p$ in terms of…

Rook theory has been investigated by many people since its introduction by Kaplansky and Riordan in 1946. Goldman, Joichi and White in 1975 showed that the sum over $k$ of the product of the $(n-k)$-th rook numbers multiplied by the $k$-th…

Let $G$ be a connected simply-connected simple algebraic group over $\mathbb{C}$ and let $T$ be a maximal torus, $B\supset T$ a Borel subgroup and $K$ a maximal compact subgroup. Then, the product in the (algebraic) based loop group…

We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation…

The rim-hook rule for quantum cohomology of the Grassmannian allows one to reduce quantum calculations to classical calculations in the cohomology of the Grassmannian. We use the Abelian/non-Abelian correspondence for cohomology to prove a…

The quantum cohomology of Grassmannians exhibits two symmetries related to the quantum product, namely a \Bbb {Z}/n action and an involution related to complex conjugation. We construct a new ring by dividing out these symmetries in an…

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules…

Matrices over a finite field having fixed rank and restricted support are a natural $q$-analogue of rook placements on a board. We develop this $q$-rook theory by defining a corresponding analogue of the hit numbers. Using tools from coding…

We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by…

The main result of this announcement is a formula for the tensor product of the class of a homogeneous line bundle with a Schubert class, expressed as a K(X)-linear combination of Schubert classes. We believe that this formula is the most…

We compute the additive structure of the Hermitian $K$-theory spectrum of an even-dimensional Grassmannian over a base field $k$ of characteristic zero in terms of the Hermitian $K$-theory of $X$, using certain symmetries on Young diagrams.…

We prove a conjecture of Buch and Mihalcea in the case of the incidence variety X=Fl(1,n-1;n) and determine the structure of its (T-equivariant) quantum K-theory ring. Our results are an interplay between geometry and combinatorics. The…

We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Moebius inversion. We give a direct…

We provide a geometric interpretation for the connecting homomorphism in the localization sequence of Hermitian $K$-theory. As an application, we compute the Hermitian $K$-theory of projective bundles and Grassmannians in the regular case.…

Let V be a symplectic vector space and LG be the Lagrangian Grassmannian which parametrizes maximal isotropic subspaces in V. We give a presentation for the (small) quantum cohomology ring QH^*(LG) and show that its multiplicative structure…

Grothendieck polynomials are important objects in the study of the $K$-theory of flag varieties. Their many remarkable properties have been studied in the context of algebraic geometry and tableaux combinatorics. We explore a new tool,…