Descent Sets, Knuth Relations and Vogan Classes

Abstract

This chapter is devoted to applications of the results of the previous Chapter 8, where we investigated the general problem of relating the (left or right) cells of W and the (left or right) cells of all its standard parabolic subgroups. Thanks to parabolic subgroups of rank 2, we recall how Knuth relations are defined. We also explain how cellular maps can be used to construct an extension of Vogan’s generalized ?-invariant.

Appendices

Notes

In 1979, Vogan [Vog1, Vog3] introduced invariants of left cells which are computable in terms of certain combinatorially defined operators $T_{\alpha \beta }$, $S_{\alpha \beta }$ where $\alpha ,\beta$ are adjacent simple roots of W. In the case where W is the symmetric group ${\mathfrak S}_n$, these invariants completely characterise the left cells [KaLu1, §5], [Vog1, §6] (see Chapter 22). Although Vogan’s invariants are not complete invariants in general, they have turned out to be extremely useful in describing left cells.

The extension of these ideas to the unequal parameter case and the use of the general setting of left cellular maps are due to Geck and the author [BoGe2] (as in the previous chapter, the easy extension to left cellular pairs is due to the author [Bon5]).

Exercises

Exercise 9.1.

(Type ${\varvec{A_3}}$). Assume here that (W, S) is of type $A_3$, that ${\mathscr {A}}={\mathbb {Z}}$ and that $\varphi (s)=1$ for all $s \in S$. Show that the coplactic classes coincide with the left Vogan $(\nabla _2,{\mathscr {R}})$-classes, so that this describes the left cells in W. Recall from Example 9.3.6 that $\nabla _2$ is the set of pairs $(I,\sigma ) \in {\mathbf L}{\mathbf C}(W, S,\varphi )$ where $|I|=2$.

Exercise 9.2

(Type ${\varvec{D_4}}$). As it is conjectured that any left cell contains an involution (see Lusztig’s Conjecture 6.5.2(L1)) and as any left cell is a union of coplactic classes (see Proposition 9.2.2), it is natural to try to check if any coplactic class contains an involution. Show that this is not the case when (W, S) is of type $D_4$. Hint. If $S=\{s_1,s_2,s_3,t\}$ with $m_{s_it}=3$ and $m_{s_is_j}=2$, then check that $\{s_1s_2s_3ts_2,ts_1s_2s_3ts_2\}$ is a coplactic class containing no involution.

Remark.

If (W, S) is of type $D_4$, there are in fact 8 coplactic classes in W which contain no involution (six of cardinality 2, and two of cardinality 6).$\square$