Demazure crystals and tensor products of perfect Kirillov–Reshetikhin crystals with various levels

Abstract

In this paper, we study a tensor product of perfect Kirillov-Reshetikhin crystals (KR crystals for short) whose levels are not necessarily equal. We show that, by tensoring with a certain highest weight element, such a crystal becomes isomorphic as a full subgraph to a certain disjoint union of Demazure crystals contained in a tensor product of highest weight crystals. Moreover, we show that this isomorphism preserves their Z-gradings, where the Z-grading on the tensor product of KR crystals is given by the energy function, and that on the other side is given by the minus of the action of the degree operator.

Randomized box–ball systems, limit shape of rigged configurations and thermodynamic Bethe ansatz

“…The Fermionic form M (B, λ, w) is defined by the same formulas as (30)- (35). Theorem 4.2 11 is valid for D (1) n [34,31] and conjecturally valid for E…”

A Uniform Model for Kirillov-Reshetikhin Crystals I: Lifting the Parabolic Quantum Bruhat Graph

“…The overlapping cases are the simply-laced affine root systems A (1) n , D (1) n and E (1) 6,7, 8 . It is known [8,9,11,17,18,36,37,43] that certain affine Demazure characters (including those for the simply-laced affine root systems) can be expressed in terms of KR crystals, which motivates the relation between P and X. For types A (1) n and C (1) n , the above mentioned relation between P and X was achieved in [21,25] by establishing a combinatorial formula for the Macdonald polynomials at t = 0 from the Ram-Yip formula [40], and by using explicit models for the one-column KR crystals [10].…”

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

“…the perfectness conjecture). We say that the above tensor product B := N j=1 B r j ,s j is of level bounded by ℓ if ℓ is such that ⌈s j /c r j ⌉ ≤ ℓ for all j. Theorem 2.6 was generalized in [Nao13,Prop. 5.16] in the following way: when B (still of nonexceptional type) is a tensor product of level bounded by ℓ, and s j /c r j ∈ Z, then B ⊗ u ℓΛ 0 is isomorphic to a direct sum of Demazure crystals.…”

“…When the (nonexceptional type) KR crystals in the tensor product are perfect of mixed levels, Naoi showed that one obtains a direct sum of Demazure crystals [Nao13]. In addition, this relationship was given for B r,1 in all types [Kas05] and can be extended to tensor products by the techniques of [Nao13] (see also [Nao12] for an alternative proof). Further connections of KR crystals, viewed as classical crystals, were described in [FL06,FL07].…”

“…8.1] (which was essentially proved in [KKM + 92a]). The claim holds for all affine types since [ST12, Lemma 7.3] holds, by using the general definition of energy [KKM + 92a, HKO + 99, HKO + 02]; see[Nao13, Lemma 6.4] as well. We also note that there is no assumption of perfectness.…”