Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userTensor product decomposition theorem for quantum Lakshmibai-Seshadri paths and standard monomial theory for semi-infinite Lakshmibai-Seshadri paths
217
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Let $lambda$ be a (level-zero) dominant integral weight for an untwisted affine Lie algebra, and let $mathrm{QLS}(lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = infty$ of the nonsymmetric Macdonald polynomial $E_{w lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $lambda$, $mu$, we have an isomorphism $Theta : mathrm{QLS}(lambda + mu) rightarrow mathrm{QLS}(lambda) otimes mathrm{QLS}(mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
rate research
Read More
We give a simple crystal theoretic interpretation of the Lascouxs expansion of a non-symmetric Cauchy kernel $prod_{i+ jleq n+1}(1-x_iy_j)^{-1}$, which is given in terms of Demazure characters and atoms. We give a bijective proof of the non-symmetric Cauchy identity using the crystal of Lakshmibai-Seshadri paths, and extend it to the case of continuous crystals.
We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose tangent bundle has positive Seshadri constant at a point is isomorphic to a projective space. We conjecture that the Fano condition can be removed. Among other results in this direction, we prove the conjecture for surfaces. Second, we restate a classical conjecture on the nef cone of self-products of curves in terms of semistability of higher conormal sheaves, which we use to identify new nef classes on self-products of curves. Third, we prove that our Seshadri constants can be used to control separation of jets for direct images of pluricanonical bundles, in the spirit of a relative Fujita-type conjecture of Popa and Schnell.
We develop a local positivity theory for movable curves on projective varieties similar to the classical Seshadri constants of nef divisors. We give analogues of the Seshadri ampleness criterion, of a characterization of the augmented base locus of a big and nef divisor, and of the interpretation of Seshadri constants as an asymptotic measure of jet separation. We also study the case of arbitrary codimension.
We propose a definition of equivariant (with respect to an Iwahori subgroup) $K$-theory of the formal power series model $mathbf{Q}_{G}$ of semi-infinite flag manifold and prove the Pieri-Chevalley formula, which describes the product, in the $K$-theory of $mathbf{Q}_{G}$, of the structure sheaf of a semi-infinite Schubert variety with a line bundle (associated to a dominant integral weight) over $mathbf{Q}_{G}$. In order to achieve this, we provide a number of fundamental results on $mathbf{Q}_{G}$ and its Schubert subvarieties including the Borel-Weil-Bott theory, whose special case is conjectured in [A. Braverman and M. Finkelberg, Weyl modules and $q$-Whittaker functions, Math. Ann. 359 (2014), 45--59]. One more ingredient of this paper besides the geometric results above is (a combinatorial version of) standard monomial theory for level-zero extremal weight modules over quantum affine algebras, which is described in terms of semi-infinite Lakshmibai-Seshadri paths. In fact, in our Pieri-Chevalley formula, the positivity of structure coefficients is proved by giving an explicit representation-theoretic meaning through semi-infinite Lakshmibai-Seshadri paths.
We prove a Pieri-Chevalley formula for anti-dominant weights and also a Monk formula in the torus-equivariant $K$-group of the formal power series model of semi-infinite flag manifolds, both of which are described explicitly in terms of semi-infinite Lakshmibai-Seshadri paths (or, equivalently, quantum Lakshmibai-Seshadri paths). In view of recent results of Kato, these formulas give an explicit description of the structure constants for the Pontryagin product in the torus-equivariant $K$-group of affine Grassmannians and that for the quantum multiplication of the torus-equivariant (small) quantum $K$-group of finite-dimensional flag manifolds. Our proof of these formulas is based on standard monomial theory for semi-infinite Lakshmibai-Seshadri paths, which is established in our previous work, and also uses a string property of Demazure-like subsets of the crystal basis of a level-zero extremal weight module over a quantum affine algebra.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
