[Paper Review] Fusion products of Kirillov-Reshetikhin modules and the X = M conjecture

Katsuyuki Naoi|arXiv (Cornell University)|Sep 12, 2011

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper establishes the X = M conjecture for affine Lie algebras of type $A_n^{(1)}$ and $D_n^{(1)}$ by proving a fermionic form identity involving fusion products of Kirillov-Reshetikhin (KR) modules and Demazure operators. Using Joseph functors, it constructs the fusion product from one-dimensional modules and shows that the character of the fusion product matches the fermionic form via Demazure operators, thereby confirming the conjecture in these cases.

ABSTRACT

In this article, we show in the ADE case that the fusion product of Kirillov-Reshetikhin modules for a current algebra, whose character is expressed in terms of fermionic forms, can be constructed from one-dimensional modules by using Joseph functors. As a consequence, we obtain some identity between fermionic forms and Demazure operators. Since the same identity is also known to hold for one-dimensional sums of nonexceptional type, we can show from these results the X = M conjecture for type $A_n^{(1)}$ and $D_n^{(1)}$.

Motivation & Objective

• To prove the X = M conjecture for affine Lie algebras of type $A_n^{(1)}$ and $D_n^{(1)}$.
• To establish a connection between fermionic forms and Demazure operators via fusion products of Kirillov-Reshetikhin modules.
• To show that the fusion product of KR modules for the current algebra ${\mathfrak{g}}_0 \otimes \mathbb{C}[t]$ can be constructed from one-dimensional modules using Joseph functors.

Proposed method

• Construct the fusion product of Kirillov-Reshetikhin modules for the current algebra ${\mathfrak{g}}_0 \otimes \mathbb{C}[t]$ using generators and relations.
• Apply Joseph functors to realize the fusion product as a quotient of a module induced from one-dimensional modules.
• Use Demazure operators to express the character of the fusion product in terms of exponential weights and Weyl group actions.
• Leverage the fact that the fusion product has a Demazure flag to relate its character to the action of Demazure operators on the weight lattice.
• Prove that the character of the fusion product matches the fermionic form via an identity involving $D_{t_{w_0(\varpi_{r_j})}}$ operators.
• Establish the isomorphism between the fusion product and a module defined by iterated Demazure operators, thereby proving the key identity.

Experimental results

Research questions

• RQ1 Does the fermionic form of the one-dimensional sum of KR crystals for $A_n^{(1)}$ and $D_n^{(1)}$ match the character of a module defined by Demazure operators?
• RQ2 Can the fusion product of Kirillov-Reshetikhin modules be realized as a quotient of a module built from one-dimensional modules via Joseph functors?
• RQ3 Is there a canonical isomorphism between the fusion product of KR modules and a module constructed via iterated Demazure operators in the $ADE$ case?
• RQ4 Does the identity between one-dimensional sums and Demazure operators extend to the fermionic form in the $A_n^{(1)}$ and $D_n^{(1)}$ cases?
• RQ5 Can the X = M conjecture be fully confirmed for $A_n^{(1)}$ and $D_n^{(1)}$ using this fusion product and Demazure operator construction?

Key findings

• The X = M conjecture is confirmed for affine Lie algebras of type $A_n^{(1)}$ and $D_n^{(1)}$ by proving the identity between fermionic forms and Demazure operators.
• The fusion product of Kirillov-Reshetikhin modules for ${\mathfrak{g}}_0 \otimes \mathbb{C}[t]$ is isomorphic to a module constructed via iterated Demazure operators.
• The character of the fusion product is shown to equal $D_{t_{w_0(\varpi_{r_p})}} \cdots D_{t_{w_0(\varpi_{r_1})}}(e^{\ell_1\Lambda_0})$ up to a constant factor.
• The proof relies on Joseph functors to construct the fusion product from one-dimensional modules, establishing a canonical isomorphism.
• The Demazure operator identity holds for the fermionic form, extending previous results from one-dimensional sums to the full fermionic expression.
• The result implies that the X = M conjecture holds for perfect KR crystals in types $B_n^{(1)}$ and $C_n^{(1)}$ if Joseph's theorem extends to non-simply laced types.