NEW ZEALAND JOURNAL OF MATHEMATICS Volume 26 (1997), 151-153

A NEW RELATION ON A WEYL GROUP

S a m y A. Y o u s s e f a n d S .G . H u l s u r k a r

(Received December 1995)

Abstract. A new partial order on a Weyl group was introduced earlier. We define here a relation on W using that partial order and derive some results. We prove a necessary and sufficient condition for such a relation to exist for given two elements of the Weyl group in a special case. The general criteria seems to be difficult to find as these discussions involve the underlying root system. These results have been used successfuly elsewhere to study a graph structure on a Weyl group.

1. Introduction

We have introduced a graph structure on Weyl groups and investigated the usual graph theory problems elsewhere. Among the important problems are planarity[1], connectivity [2], graph automorphism [3] and the girth [4] of such graphs. The reason for introducing this graph structure on Weyl groups is the new partial order defined on the Weyl groups in the proof of the Verma’s conjecture on Weyl’s dimension polynomial [5]. This partial order gives a graph structure on Weyl groups in a very natural manner. The incidence matrix of this graph has applications in the representations of Chevalley groups [6]. In this paper we give two basic results on the relation on a Weyl group which is fundamental for the graphs on Weyl groups.

2. Basic Definitions

We use the notations and definitions given in [5]. We refer to [7] for details of the root system and the Weyl groups. Here we describe the relevant concepts very briefly.

Let $ be a root system in a real vector space E of dimension n with positive definite inner product ( , ). For a € <i>, let Ra denote the reflection given by ARa = A - (A, q:v)q: for A € E, where a v = 2 a /(a ,a ). The Weyl group W of $ is generated by Ra,a £ Let - ,a n be the simple roots of $and Ai,A2, . . . ,An be the corresponding fundamental weights, i.e., they are de­fined by (Ai,ctj) = Sij (Kronecker delta). Then Weyl group W is generated by R i,R 2, . .. , Rn where Ri = Rai,* = 1 ,2 ,... , n. We can express any element of W as a product of these generators. For a € W , define the length of <r, £(&), as the minimum length of an expression for a in terms of the generators Ri, R 2, . .. ,R n- Let <Jo be the unique element of W with maximum length. For a € W , define tip = Yhiei where Ia = {i \ 1 < i < n,£(crRi) < ^(cr)}. Then ea is defined by ea = 6a (j~x. Let 8 = ^i-

1991 AMS Mathematics Subject Classification: Primary 20F55; Secondary 05C25.

152 SAMY A. YOUSSEF AND S.G. HULSURKAR

A point A € E is called W-regular if A lies in the interior of a Weyl chamber. This is also equivalent to D (A) ^ 0 where D (A) is the Weyl’s dimension polynomial. For cr, r e W , define a —» r iff —eaaQ + s T is VT-regular. It is shown in [5] that only one of — eaao + eT and —sT(7q 4 ea is W-regular which means only one of a —► r and r —* cr holds. A graph r(VF(<I>)) on a Weyl group is defined where vertices are elements of the Weyl group W and (cr, r) is an edge in it if (cr, r) is an unordered pair with cr ^ r and either a —* r or r —> a. This definition of a graph on a Weyl group depends upon the relation —> defined on W . As mentioned earlier this graph structure has been studied elsewhere [8]. Here we give some results on the relation —> on a Weyl group W .

3. Some Theorems on New Relation

We give below two theorems regarding the new relation —* on W . The first theo­rem gives the necessary condition that for a, t € W the relation a —> r holds and the second theorem gives a necessary and sufficient condition in special circumstances.

Theorem 3.1. Let a ,r € W and a —> r. Then ST — 8a = S' 4 ^2 a where 8' is either zero or a sum of the positive A* ’s and ^ c t is either zero or a sum of the positive roots.

Proof. The assumption cr —> r implies that —eaaQ 4 eT is VT-regular,i.e., —eaao 4 £t = {8 4 8')p~l for some p € W and S' is either zero or a sum of positive Aj’s. Therefore, 6Tr ~ l -I- (6 — S a ) ^ 1 = (8 4- i.e., STr ~ 1p 4(8 — <5a)cr-1 p = S 4- S'. If A is dominant then for all a € W , Act < A, (see [7], pg. 68, Lemma A), i.e., A — A a is zero or a sum of positive roots . This gives 8r — yi Q- + (8 — 8a) — ^2 ft = 8 4 S', i.e., 6t — Sa = 8' 4 ol' -I- 0' = S' 4 a. where ]T) a is the sum of the positive roots. □

Let J be a subset of / = {1 ,2 ,... , n}. The subgroup W j generated by R j ,j e J is again a Weyl group corresponding to the root system whose simple roots are a j ,j e J. The root system 3>j need not be irreducible . Note that W = Wj. AlsoI — J denotes the compliment of J in I.

For cr,T € W with Sa = Sr , we have the following theorem in terms of the group structure of W alone.

Theorem 3.2. Let a ,r € W and Ia = IT. Then a —> r iff a = t u v , where u e W i - i g and v G W ia.

Proof. First note that Ia — IT implies Sa = ST. Then by the definition of I<j,8crUi — Sa iff Ui € W ! - U or also (6 — Sff)vi = 8 — Sa iff v\ € W ia. These claims follow essentially from A * Rj = A* for j ^ i.

Let cr = t u v where u G W i - i a and v £ W ja. Then — ea(To 4 sT = {8 — 6CT)cr-1 4 ST r _1 = (8 — 8a) v ~ x u_1 t - 1 4 ST r -1 = ((5 — <5CT)i>-1 4 8au) u~1 r -1 = (6 — Sa 4 8a) u~x t ~ 1 = 8 u~l T~l . This shows that - e aa0 + £t is W -̂regular, i.e., cr —> r.

A NEW RELATION ON A W EYL GROUP 153

Conversely, let a —* r. Suppose + eT = (8 — 8a)a 1 + 8t t 1 = (6 + 8')p 1where p e W and 6' is zero or a sum of positive A*’s. Therefore,

(<5 - 8a) a ~ l p + 8r t _ 1 p = 6 + 8'. (1)

Thus we can assume ([7], pg. 68, Lemma A),

(<5 - <5ff) <t_1 p = ( 8 - 8 a) and 8T r _1 p = 8T - (2)

where J2a an ̂ J2 0 are zero or sums of positive roots. Substituting (2) in (1), we get (8 — Scr) — ^ a + ST — (3 = 6 + S' which reduces to 5' = — Y^a — since ba = 6T. But if 6' is the sum of positive Ai’s then it cannot be the sum of negative roots . Therefore 6 ' = = = 0- From (2), (<5 — 6cr)a~1p = 6 — 8a and 8t t ~ 1 p = ST. This shows that a ~l p € W ia and r _1 p e by theobservations made in the beginning of the proof. Suppose cr-1 p = v ~ l e W ja and r -1 p = u € W i - j a. This simplifies to a = pv and p = t u . Therefore a = ruv where u € W i - i a and v e W ia. □

References

1. Samy A. Youssef and S.G. Hulsurkar, On planarity of graphs on Weyl groups Tamkang Journal of Mathematics 26 (1995), 361 - 369.

2. Samy A. Youssef and S.G. Hulsurkar, On connectedness of graphs on direct product of Weyl groups, Archivum Mathematicum 31 (1995), 299-304.

3. Samy A. Youssef and S.G. Hulsurkar, On automorphisms of graphs on Weyl groups, Journal of Mathematical and Physical Sciences 26 (1992), 419 - 423.

4. Samy A. Youssef and S.G. Hulsurkar, More on the girth of graphs on Weyl groups, Archivum Mathematicum 29 (1993), 19-23.

5. S.G. Hulsurkar, Proof of Verma’s conjecture on W eyl’s dimension polynomial, Inventiones Math. 27 (1974), 45-52.

6. L. Chastkofsky, Variation on Hulsurkar’s matrix with applications to represena- tions of algebraic Chevalley groups, J. Algebra 82 (1983), 255-274.

7. J.E. Humphreys, Introduction to Lie Algebras and Representations Theory, Springer-Verlag, New York, 1972.

8. Samy A. Youssef, Graphs on Weyl groups, Ph. D. Thesis, Indian Institute of Technology, Kharagpur, 1993.

Samy A. Youssef and S.G. Hulsurkar Department of Mathematics Indian Institute of Technology Kharagpur - 721 302 INDIAsghulsur@maths.iitkgp.emet.in