[Paper Review] Virtual crystals and fermionic formulas of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$

Masato Okado, Anne Schilling+1 more|ArXiv.org|May 2, 2001

Algebraic structures and combinatorial models25 references3 citations

TL;DR

This paper introduces virtual crystals for affine types $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$ by extending embeddings of classical crystals into type $A_{2n-1}$, conjecturing they realize crystal bases of finite-dimensional $U_q'(\mathfrak{g})$-modules. It proves fermionic formulas for one-dimensional configuration sums in specific cases using rigged configurations and contragredient duality, establishing a link between virtual crystals and known fermionic expressions.

ABSTRACT

We introduce ``virtual'' crystals of the affine types $g=D_{n+1}^{(2)}$, $A_{2n}^{(2)}$ and $C_n^{(1)}$ by naturally extending embeddings of crystals of types $B_n$ and $C_n$ into crystals of type $A_{2n-1}$. Conjecturally, these virtual crystals are the crystal bases of finite dimensional $U_q'(g)$-modules associated with multiples of fundamental weights. We provide evidence and in some cases proofs of this conjecture. Recently, fermionic formulas for the one dimensional configuration sums associated with tensor products of the finite dimensional $U_q'(g)$-modules were conjectured by Hatayama et al. We provide proofs of these conjectures in specific cases by exploiting duality properties of crystals and rigged configuration techniques. For type $A_{2n}^{(2)}$ we also conjecture a new fermionic formula coming from a different labeling of the Dynkin diagram.

Motivation & Objective

• To construct virtual crystals for affine types $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$ by extending classical crystal embeddings into type $A_{2n-1}$.
• To conjecture that these virtual crystals realize the crystal bases of finite-dimensional $U_q'(\mathfrak{g})$-modules associated with multiples of fundamental weights.
• To prove fermionic formulas for one-dimensional configuration sums in specific cases using rigged configuration techniques and duality.
• To provide evidence for the conjecture by verifying classical decomposition and structural properties of virtual crystals.

Proposed method

• Extend known embeddings of $B_n$ and $C_n$ crystals into $A_{2n-1}$ crystals to finite affine crystals, defining virtual crystals $V^{r,s}$.
• Use rigged configuration theory to analyze the one-dimensional configuration sums of virtual crystals.
• Apply contragredient duality to relate elements in virtual crystals to their duals, ensuring compatibility with fermionic formulas.
• Employ inverse algorithms on rigged configurations to reconstruct tensor product components and verify tableau conditions.
• Use the action of $j_{r,1}^{-1} \circ \cdots$ and $i_{r,1}^{-1} \circ \cdots$ to remove singular strings and reduce to base cases.
• Verify that the resulting virtual crystals satisfy necessary conditions for being in the image of the rigged configuration bijection.

Experimental results

Research questions

• RQ1 Do the virtual crystals $V^{r,s}$ for types $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$ realize the crystal bases of finite-dimensional $U_q'(\mathfrak{g})$-modules?
• RQ2 Can the one-dimensional configuration sums of virtual crystals be expressed as fermionic formulas?
• RQ3 Is the classical decomposition of virtual crystals $V^{r,s}$ consistent with the expected decomposition into irreducible $A_{2n-1}$-crystals?
• RQ4 Does contragredient duality hold for the rigged configurations associated with virtual crystals?
• RQ5 Can the virtual crystal $V^{r,1}$ be explicitly characterized to enable comparison with fermionic formulas?

Key findings

• The virtual crystals $V^{r,s}$ for types $A_{2n}^{(2)}$, $A_{2n}^{(2)lat}$, and $C_n^{(1)}$ are proven to be isomorphic to the conjectured crystals $B^{r,s}$ for $s=1$.
• For type $D_{n+1}^{(2)}$, the virtual crystals $V^{r,s}$ are shown to have at least the expected classical components for all $s \geq 1$.
• The one-dimensional configuration sum of $V^{r,1}$ matches the fermionic formula conjectured by Hatayama et al. in the case of $A_{2n}^{(2)}$.
• The rigged configuration map $\overline{\phi} \circ \Psi_R$ is shown to be surjective onto the image of the virtual crystal, confirming compatibility with fermionic expressions.
• The condition that $(uv)|_{[n]}$ is a tableau is satisfied due to contragredient duality and the derived inequality $u_{a+i-c} \leq n-c$, which prevents non-tableau configurations.
• The proof establishes that $|u|_{[n]} - |v|_{[n]}| = n - r$, confirming the correct weight condition for the virtual crystal components.