[Paper Review] Young tableaux and crystal $B(\infty)$ for finite simple Lie algebras

Jin Hong, Hyeonmi Lee|arXiv (Cornell University)|Jul 21, 2005

Algebraic structures and combinatorial models9 references4 citations

TL;DR

This paper provides an explicit realization of the crystal $×frak{B}(\infty)$, the crystal base of the negative part of quantum groups, using large semi-standard Young tableaux for finite simple Lie algebras of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$. It constructs a crystal structure on equivalence classes of these tableaux and proves isomorphism to $×frak{B}(\infty)$, offering a new tableau-based model that generalizes previous work and connects to Cliff's realization via an explicit isomorphism.

ABSTRACT

We study the crystal base of the negative part of a quantum group. An explicit realization of the crystal is given in terms of Young tableaux for types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$. Connection between our realization and a previous realization of Cliff is also given.

Motivation & Objective

• To provide an explicit, combinatorial realization of the crystal $×frak{B}(\infty)$ for finite simple Lie algebras of types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$ using Young tableaux.
• To extend the known tableau realization of highest weight crystals $×frak{B}(\lambda)$ to the infinite-dimensional case $×frak{B}(\infty)$.
• To establish an isomorphism between the new tableau-based realization and Cliff's earlier realization of $×frak{B}(\infty)$.
• To provide a uniform framework for constructing Nakajima monomial realizations of $×frak{B}(\infty)$ in types $B_n$, $C_n$, $D_n$, and $G_2$, analogous to the known $A_n$ case.

Proposed method

• Define 'large' semi-standard Young tableaux as those with sufficient trailing zeros in each row to allow for Kashiwara operator actions without loss of largeness.
• Introduce an equivalence relation on large tableaux where two tableaux are equivalent if, for each $i$ and $j \succ i$, the number of $j$-boxes in the $i$-th row is identical.
• Construct the set $\mathcal{T}(\infty) = \mathcal{T}^L / \sim$ as the set of equivalence classes of large tableaux.
• Define Kashiwara operators $\tilde{f}_i$ and $\tilde{e}_i$ on $\mathcal{T}(\infty)$ by lifting actions from representatives in $\mathcal{T}^L$, ensuring well-definedness via the equivalence relation.
• Prove that the resulting crystal $\mathcal{T}(\infty)$ satisfies the axioms of a crystal, including the signature rule and the connectedness of the crystal graph.
• Establish an explicit isomorphism between $\mathcal{T}(\infty)$ and Cliff's realization of $\mathcal{B}(\infty)$ by matching tableaux and their corresponding monomial expressions.

Experimental results

Research questions

• RQ1 Can the crystal $\mathcal{B}(\infty)$ for finite simple Lie algebras be realized combinatorially using Young tableaux beyond type $A_n$?
• RQ2 Is there a uniform construction of $\mathcal{B}(\infty)$ using large semi-standard tableaux that respects the crystal structure across types $A_n$, $B_n$, $C_n$, $D_n$, and $G_2$?
• RQ3 How does the new tableau realization of $\mathcal{B}(\infty)$ relate to Cliff's earlier realization based on different combinatorial objects?
• RQ4 Can the new tableau model be used to construct Nakajima monomial realizations of $\mathcal{B}(\infty)$ in types $B_n$, $C_n$, $D_n$, and $G_2$?

Key findings

• The paper constructs a new realization of $\mathcal{B}(\infty)$ as the set of equivalence classes of large semi-standard Young tableaux under a specific equivalence relation based on row counts of labels greater than $i$ in the $i$-th row.
• The crystal structure on $\mathcal{T}(\infty)$ is well-defined and isomorphic to the standard $\mathcal{B}(\infty)$ crystal, as verified by checking the Kashiwara operator axioms and connectedness.
• For type $A_n$, the new realization is equivalent to the one previously obtained by [hm], though derived through a distinct method based on tableau largeness and equivalence classes.
• An explicit isomorphism is constructed between the new tableau realization and Cliff's realization of $\mathcal{B}(\infty)$, showing that the two combinatorial models are equivalent.
• The framework provides a foundation for constructing Nakajima monomial realizations of $\mathcal{B}(\infty)$ in types $B_n$, $C_n$, $D_n$, and $G_2$, extending the known $A_n$ case.