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Join operation for the Bruhat order and Verma modules

Hankyung Ko, Volodymyr Mazorchuk, Rafael Mrđen

Abstract

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type $A$. The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.

Join operation for the Bruhat order and Verma modules

Abstract

We observe that the join operation for the Bruhat order on a Weyl group agrees with the intersections of Verma modules in type . The statement is not true in other types, and we propose a conjectural statement of a weaker correspondence. Namely, we introduce distinguished subsets of the Weyl group on which the join operation conjecturally agrees with the intersections of Verma modules. We also relate our conjecture with a statement about the socles of the cokernels of inclusions between Verma modules. The latter determines the first Ext spaces between a simple module and a Verma module. We give a conjectural complete description of such socles, which we verify in a number of cases. Along the way, we determine the poset structure of the join-irreducible elements in Weyl groups and obtain closed formulae for certain families of Kazhdan-Lusztig polynomials.

Paper Structure

This paper contains 33 sections, 75 theorems, 128 equations, 10 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hecke algebra and Kazhdan-Lusztig basis
  4. The small and penultimate KL cells
  5. Kazhdan-Lusztig combinatorics, descent sets, and Verma modules
  6. Join-irreducible elements and dissectors
  7. Join operation on Bruhat orders
  8. Join operation and Verma modules
  9. Intersection conjecture
  10. Socle-sum property and the intersection conjecture
  11. Another interpretation of Question \ref{['con2']}
  12. Half socle-sum property via combinatorics
  13. Socle-sum property and $\mathbf{JM}"$
  14. Join-irreducibles and Verma modules
  15. Strategies
  16. ...and 18 more sections

Key Result

Theorem B

Suppose $\mathfrak{g} = \mathfrak{sl}_n$. Then for any $U\subseteq W$, we have $\bigvee U =w\in W$ if and only if $\bigcap_{x\in U}\Delta_x = \Delta_w$.

Figures (10)

  • Figure 1: Bruhat graph of the penultimate two-sided cell in type $B_{n+1}$.
  • Figure 2: Composition factors of $\Delta_e$ from the penultimate cell, for $n = 1, 2, 3, 4$.
  • Figure 3: The octahedron for $n = 1, 2, 3$.
  • Figure 4: Bruhat graph of the penultimate two-sided cell in type $D_{n+2}$ with $n$ even.
  • Figure 5: The Bruhat graph of ${}_{i}\mathbf{JI}_{j}$ in types $B_{n+1}$ (left) and $D_{n+2}$ (right), for $1 \leq i \leq j \leq n$ and $i \leq n-j$, with solid arrows socle-killing. In the right-hand side diagram, all but the first two arrows from (\ref{['al:D_OAX']}) are omitted, as well as all but the last four from (\ref{['al:D_OBX']}). The case $i > n-j$ is similar, but without type $O_B$ elements (i.e., no dashed arrows).
  • ...and 5 more figures

Theorems & Definitions (150)

  • Theorem B
  • Conjecture C
  • Definition D
  • Theorem E
  • Theorem F
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Proposition 2.4
  • ...and 140 more