On higher level Kirillov–Reshetikhin crystals, Demazure crystals, and related uniform models

Abstract

We show that a tensor product of nonexceptional type Kirillov-Reshetikhin (KR) crystals is isomorphic to a direct sum of Demazure crystals; we do this in the mixed level case and without the perfectness assumption, thus generalizing a result of Naoi. We use this result to show that, given two tensor products of such KR crystals with the same maximal weight, after removing certain 0-arrows, the two connected components containing the minimal/maximal elements are isomorphic. Based on the latter fact, we reduce a tensor product of higher level perfect KR crystals to one of single-column KR crystals, which allows us to use the uniform models available in the literature in the latter case. We also use our results to give a combinatorial interpretation of the Q-system relations. Our results are conjectured to extend to the exceptional types.

Simplified presentations and embeddings of Demazure modules

“…If B r,s denotes the KR crystal of classical highest weight s̟ r and B = N j=1 B r j ,s j is a tensor product of KR crystals bounded by k, i.e. ⌈ s j cr j ⌉ ≤ k for all 1 ≤ j ≤ N (see [40] for the table defining the integers c r ), then the proof of [33,Theorem 3.8] shows for nonexceptional types that the Demazure crystal of D k µ for anti-dominant µ = N j=1 r j w 0 ̟ j can be obtained by the following steps.…”

“…It is known that the Demazure module V w (Λ) admits a crystal base [27] which is the full subgraph of the crystal base of the irreducible integrable highest weight module whose vertices consist precisely of those elements that are reachable by raising operators from the unique element of weight wΛ. There is a connection to the tensor product of Kirillov-Reshetikhin crystals which has been worked out by various authors in the last years (see for example [33,37,40] and the references therein). The main result of [33], which seems to be the most general version so far, states that the g-stable level k Demazure module for non-exceptional types appear inside a tensor product of Kirillov-Reshetikhin crystals by removing all edges that are not level k Demazure edges (see [33,Section 2] for a precise definition) and picking the connected component which contains the unique element of extremal weight wΛ.…”

“…There is a connection to the tensor product of Kirillov-Reshetikhin crystals which has been worked out by various authors in the last years (see for example [33,37,40] and the references therein). The main result of [33], which seems to be the most general version so far, states that the g-stable level k Demazure module for non-exceptional types appear inside a tensor product of Kirillov-Reshetikhin crystals by removing all edges that are not level k Demazure edges (see [33,Section 2] for a precise definition) and picking the connected component which contains the unique element of extremal weight wΛ. Finding very explicit combinatorial models for these crystals is still a very important and open problem.…”

“…The crystal theoretic approach to Demazure modules and the realization inside a tensor product obtained in [33] is one motivation of the article to embed any higher level Demazure crystal (no restrictions on the type nor on the extremal weight) into a tensor product of Demazure crystals which are in some way easier to understand. We attack this problem algebraically which we will explain now in more details.…”

“…Finally we propose, similar to the approach taken in [33], a way to come up with classical decompositions. For any weight µ ∈ P we define a subset R − (µ) (see Section 2) and if this set is non-empty we can find a maximal semi-simple subalgebra g 0 ⊆ g corresponding to the nodes in R − (µ); the g-stable case corresponds to R + = R − (µ).…”