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Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)

Ben Salisbury, Adam Schultze, Peter Tingley

TL;DR

The paper studies $B(\infty)$ for ADE Lie algebras through Lusztig's PBW bases realized on Kostant partitions, aiming to simplify crystal operator actions by identifying reduced expressions that admit bracketing rules. It establishes $i$-semi-adapted reduced expressions in all simply-laced types except $E_8$, and provides explicit bracketing descriptions for types $A$ and $D$, including a new type $D_n$ isomorphism with marginally large tableaux and a note on $*$-operators. It connects these PBW realizations to standard tableaux combinatorics, showing how multisegments in type $A$ and a new D-type marginally large tableaux correspondence realize $B(\infty)$ and interact with embeddings of finite crystals. The results offer concrete, combinatorial means to compute crystal operators and to understand embeddings $B(\lambda) \hookrightarrow B(\infty)$, while outlining conjectures and partial results for $E$-types and non-simply-laced types.

Abstract

Lusztig's theory of PBW bases gives a way to realize the infinity crystal for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except $E_8$, and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.

Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)

TL;DR

The paper studies for ADE Lie algebras through Lusztig's PBW bases realized on Kostant partitions, aiming to simplify crystal operator actions by identifying reduced expressions that admit bracketing rules. It establishes -semi-adapted reduced expressions in all simply-laced types except , and provides explicit bracketing descriptions for types and , including a new type isomorphism with marginally large tableaux and a note on -operators. It connects these PBW realizations to standard tableaux combinatorics, showing how multisegments in type and a new D-type marginally large tableaux correspondence realize and interact with embeddings of finite crystals. The results offer concrete, combinatorial means to compute crystal operators and to understand embeddings , while outlining conjectures and partial results for -types and non-simply-laced types.

Abstract

Lusztig's theory of PBW bases gives a way to realize the infinity crystal for any simple complex Lie algebra where the underlying set consists of Kostant partitions. In fact, there are many different such realizations, one for each reduced expression for the longest element of the Weyl group. There is an algorithm to calculate the actions of the crystal operators, but it can be quite complicated. For ADE types, we give conditions on the reduced expression which ensure that the corresponding crystal operators are given by simple combinatorial bracketing rules. We then give at least one reduced expression satisfying our conditions in every type except , and discuss the resulting combinatorics. Finally, we describe the relationship with more standard tableaux combinatorics in types A and D.

Paper Structure

This paper contains 17 sections, 10 theorems, 24 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. General
  4. Crystals
  5. Reduced expressions, convex orders, and PBW bases
  6. Semi-adapted words and bracketing
  7. Semi-adapted words
  8. Bracketing rules
  9. Existence of semi-adapted words
  10. Explicit descriptions
  11. Type $A$
  12. Type $D$
  13. Types $E_6$, $E_7$, and $E_8$
  14. Non-simply-laced types
  15. Relation to tableaux combinatorics
  16. ...and 2 more sections

Key Result

Theorem 2.3

There is a one-to-one correspondence between $R(w_0)$ and convex orders on $\Phi^+$. Explicitly, if $w_0 = s_{i_1}s_{i_2}\cdots s_{i_N}$, then the corresponding convex order $\prec$ is defined by

Figures (1)

  • Figure 5.1: The fundamental crystal of type $D_n$.

Theorems & Definitions (22)

  • Definition 2.2
  • Theorem 2.3: papi
  • Theorem 2.6: saito
  • Proposition 2.7: BZ01L10
  • Lemma 2.8: L90
  • Lemma 2.9
  • Example 2.10
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 12 more