Monoidal categorifications on twisted product of flag varieties

TL;DR

The paper constructs a monoidal categorification of the coordinate rings of twisted products of flag varieties for a simple, simply-connected, simply-laced group, unifying braid varieties and restricted double Bruhat cells. It introduces a subalgebra $\widehat{\mathcal{A}}_{v,b}$ of the bosonic extension algebra and identifies a monoidal subcategory $\mathscr{C}_{v,\beta}$ in the Hernandez–Leclerc framework whose Grothendieck ring $K_0(\mathscr{C}_{v,\beta})$ contains a cluster algebra $\mathcal{A}_0(\mathbf{s}(v,\beta))$, with cluster monomials corresponding to simple objects. The quantum version gives $K_t(\mathscr{C}_{v,\beta})\cong\widehat{\mathcal{A}}_{v,\beta}$ and an upper cluster algebra isomorphic to $\mathbb{C}[\mathring{\mathcal{Z}}_{v,\beta}]$, thereby quantizing the twisted-product coordinate rings. The work combines combinatorial seed mutations, Lusztig-parameter transitions under braid moves, and categorical constructions to provide a robust framework linking cluster structures, categorification, and geometric objects arising from flag varieties.

Abstract

For a simple, simply-connected, simply-laced algebraic group $G$, we construct a monoidal categorification of the coordinate ring of twisted products of flag varieties. This class of varieties includes, in particular, braid varieties and restricted double Bruhat cells. In addition, we define a natural subalgebra of the bosonic extension algebra $\widehat{\mathcal{A}}$ and show that this subalgebra provides a quantization of the coordinate ring of twisted products of flag varieties.

Figures (6)

• Figure 1: $Q_{\beta}$
• Figure 2: $Q_{(a,b)}$
• Figure 3: $\mu_{j^{(t-1)-}}\cdots \mu_j$
• Figure 4: $\mu_{j^{t-1}}\cdots \mu_j$
• Figure 5: $\mu_{j^{t-1}}\cdots \mu_j$

Theorems & Definitions (65)

• Theorem 1.1: Theorem \ref{['thm:categorification']}
• Example 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5
• Lemma 2.6: qin2024analogs, Lemma 6.3
• Proposition 2.7: fujita2023isomorphisms, Proposition 1.2
• Example 2.8
• Definition 2.9