Kirillov-Reshetikhin Dual Equivalence Graphs

TL;DR

This work develops Kirillov-Reshetikhin dual equivalence graphs (KR DEGs) by wiring together $0$-weight spaces of tensor products of KR crystals with generalized dual equivalence moves, extending KL DEGs to an affine-crystal setting. It proves that a natural charge- and promotion-based stratification coincides with the KR DEG-based stratification, and shows that the KR DEG for a sequence of rectangles $R$ decomposes into exactly $d_R=\gcd(m_1,...,m_\ell)$ connected components, each labeled by charge modulo $d_R$. The paper also provides a precise character theory for these components: each has a cyclic-plethysm character $\ell_{d_R}^{(i)}[s_{R_1}^{m_1/d_R}\cdots s_{R_k}^{m_k/d_R}]$, tying diagonal $GL_n$ actions on $U^{\otimes k}$ to combinatorial crystal data. Collectively, these results connect crystal combinatorics, dual equivalence frameworks, and symmetric-function plethysms, advancing understanding of Schur-positivity phenomena in KR-structured tensor powers.

Abstract

Let $U$ be a tensor product of highest weight modules of $GL_n(\mathbb C)$ corresponding to multiples of fundamental weights (i.e. rectangles). We consider three ways to stratify $U^{\otimes k}$ into components: using isotypic components of the cyclic action on tensor factors, using a generalization of the charge statistic, and using certain generalizations of Assaf's dual equivalence graphs. We conjecture that all three ways coincide, and we prove that the latter two ways coincide. The Kirillov-Reshetikhin dual equivalence graphs (KR DEGs) we introduce for this purpose are defined on $0$-weight spaces of tensor products of Kirillov-Reshetikhin crystals. They generalize Kazhdan-Lusztig dual equivalence graphs (KL DEGs) that previously appeared in the study of Kazhdan-Lusztig cells in affine type A. While the tensor products of Kirillov-Reshetikhin crystals are connected as affine crystals, the KR DEGs in general are not.

Figures (6)

• Figure 1: The above graph with only black edges is the crystal graph $B_{(2,2)}$ for $n = 3$. Adding in the blue edges, we get the corresponding KR crystal graph $B^{(2,2)}$.
• Figure 2: The KR DEG $\mathcal{T}((2^2),(1))$. The elements of the descent set of each vertex are shown in red. The edges $t_1$-$t_5$ are shown in black and the special edges $\bar{t}_1,\bar{t}_4$ are shown in blue. The edges equivalently obtained by crystal commutators $C_1$-$C_5$ are shown in orange.
• Figure 3: The KR DEG ${\mathcal{T}}((2),(1,1))$.
• Figure 4: The KR DEG $\mathcal{T}((1^3),(1^3))$ has two connected components. The elements of the descent set of each vertex are shown in red. The elements of the left and right components have charge congruent to $0$ and $1 \bmod 2$ respectively.
• Figure 5: The path from $T$ to $\mathop{\mathrm{pr}}\nolimits^{14}(T)$ for $T=\mathsf{col}(R)\in \mathcal{T}((2^2),(3^3),(3^3)).$

Theorems & Definitions (74)

• Example 2.1
• Example 2.2
• Definition 2.3
• Example 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Example 2.9
• Definition 2.10