[Paper Review] Demazure crystals and tensor products of perfect Kirillov-Reshetikhin crystals with various levels

Katsuyuki Naoi|arXiv (Cornell University)|Aug 16, 2011

Algebraic structures and combinatorial models28 references3 citations

TL;DR

This paper generalizes the isomorphism between tensor products of Kirillov-Reshetikhin (KR) crystals and Demazure crystals to cases where KR crystals have varying levels. It proves that, after tensoring with a highest weight element, such a tensor product becomes isomorphic as a full subgraph to a disjoint union of Demazure crystals inside a tensor product of highest weight crystals, preserving $η$-gradings via the energy function and degree operator.

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 gradings, where the 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.

Motivation & Objective

• To extend the known isomorphism between tensor products of perfect KR crystals (same level) and Demazure crystals to cases with different levels.
• To establish that the tensor product of KR crystals with varying levels, when tensored with a highest weight element, becomes isomorphic to a disjoint union of Demazure crystals.
• To prove that this isomorphism preserves $η$-gradings, where the grading on the KR side is given by the energy function and on the Demazure side by the negative action of the degree operator.
• To generalize results from Schilling and Tingley (2012) to non-uniform level KR crystals in nonexceptional affine Kac-Moody algebras.
• To provide a combinatorial framework using Dynkin automorphisms and translation operators to describe the structure of the resulting Demazure crystals.

Proposed method

• Utilizes the theory of Demazure crystals $B_w(\Lambda)$ as subsets of highest weight crystals $B(\Lambda)$, defined via the action of Kashiwara operators.
• Applies the concept of perfect KR crystals $B^{r, c_r\ell}$ with varying levels $\ell_j$, ensuring they are finite-dimensional and admit crystal bases.
• Introduces the use of Dynkin automorphisms $\tau$ and their induced actions on crystals to adjust weights and operators, enabling the construction of new crystal structures.
• Employs the translation operator $t_{\mu}$ to shift weights and define Demazure-type subgraphs in the tensor product of highest weight crystals.
• Constructs a nested filtration using the operators $\mathcal{F}_{w\tau}(S)$, which generate Demazure crystals via iterated application of Kashiwara operators.
• Relies on the combinatorial excellent filtration theorem to show that the target object is a disjoint union of Demazure crystals, and uses induction on the number of KR crystals to prove the isomorphism.

Experimental results

Research questions

• RQ1 Can the isomorphism between tensor products of perfect KR crystals and Demazure crystals be extended to cases where the KR crystals have different levels?
• RQ2 How can the $η$-grading on the tensor product of KR crystals (via the energy function) be related to the grading on Demazure crystals (via the degree operator)?
• RQ3 What structure emerges when a tensor product of KR crystals with varying levels is tensored with a highest weight element?
• RQ4 How do Dynkin automorphisms and translation operators contribute to constructing the isomorphic image in the highest weight crystal framework?
• RQ5 Is the resulting isomorphism still compatible with the $η$-grading when the levels are not equal?

Key findings

• The tensor product $u_{\ell_p\Lambda_0} \otimes B^{r_p,c_{r_p}\ell_p} \otimes \cdots \otimes B^{r_1,c_{r_1}\ell_1}$ is isomorphic as a full subgraph to a nested Demazure crystal construction $\mathcal{F}_{t_{\mu_p}}( \cdots \mathcal{F}_{t_{\mu_1}}(u_{\ell_1\Lambda_0}) \cdots )$.
• The isomorphism preserves the $\mathbb{Z}$-grading: the energy function on the KR side corresponds to the negative action of the degree operator on the Demazure side.
• The target object is a disjoint union of Demazure crystals, as guaranteed by the combinatorial excellent filtration theorem.
• The grading invariants are preserved: $\overline{D}(b) = D(b) + \langle \mathrm{wt}(\Psi_B(u_{\ell_p\Lambda_0} \otimes b)), d \rangle$ is constant across the isomorphism, ensuring grading compatibility.
• The proof relies on induction on the number of KR crystals and uses weight shifts via $\tau$-actions and translation operators $t_{\mu_j}$ to align the structures.
• The result generalizes Schilling and Tingley's work to non-uniform levels, showing that the Demazure crystal structure still emerges even when levels differ.