## Doctoral Theses

### Coxeter Groups and Positive Matrices.

8-28-1992

8-28-1993

#### Institute Name (Publisher)

Indian Statistical Institute

Doctoral Thesis

#### Degree Name

Doctor of Philosophy

Mathematics

#### Department

Theoretical Statistics and Mathematics Unit (TSMU-Delhi)

#### Supervisor

Bapat, Ravindra B. (TSMU-Delhi; ISI)

#### Abstract (Summary of the Work)

In this thesis, we study positive matrices (matrices whose entries are nonnegative as well as matrices which are positive semidefinite) with Coxeter groups as the underlying theme. For an exposition on Coxeter groups see Humphreys (1990).A Cozeter system consists of a pair (W, s); where W is a group and S is a set which consists of the generators of the group W. The elements of the set S have only the relations of the form (ss')m(s.) 1; where m(s, s) 1, m(s, s') = m(s,s) 2 2 for s s in S. In case no relation occurs for a pair s, s, we make the convention that m(s,s) = 00.To represent a Cozeter system (W, S) we need a finite set S of generators and a symmetric matrix M whose rows and columns are indexed by S with entries in ZU{0} subject only to the conditions : m(s,s) = 1, m(s, s) 2 2 if s + s. Equivalently, one can draw a graph G with S as the vertex set, joining vertices s and s by an edge labelled m(s, s) whenever this number ( oo allowed) is at least 3. If distinct vertices s and s are not joined, we understand that m(s, s) = 2. G is called the Cozeter graph corresponding to the Coxeter system (W, S). A Coxeter system is said to be irreducible if the corresponding Coxeter graph G is connected.The adjacency matriz of a Coxeter graph G, usually denoted by A(G) = (a,))is defined to be a square matrix of order |VI (IV] denotes the cardinality of V); where aij = 2cos(*/p) if the edge (i, j) is labelled with the integer p, and 0 if there is no edge joining the vertex i with vertex j. Note that p= oo corresponds to a, = 2 for i tj and a = 0 for i = 1,2,..,n.Since each of the generators s eS have order 2 in W, each w #1 (identity) in W can be written in the form w = 3182..s, for some s, (not necessarily distinct) in S. If r is the smallest integer for which the above expression is possible, then r is called the length of w, written I(w).The root system of W is the collection of all vectors w(a,); where w e W and SE S. To understand the root system , we look at the geometrical representation of W. Let (W, S) be the given Coxeter system. We consider a vector space V over R, having a basis (a,s e S}) which has one-to-one correspondence with S. We give a geometry on V in such a way that the 'angle' between a, and a, is iri : Based on this geometry, we define a symmetric bilinear form B on V by taking:B(a, a,) = - COS m(s, s') (This expression is interpreted to be -1 in case m(s, s) = 0o.) We then get a relation A(G) = 2(1 â€“ B).There are several criteria for the finiteness of a Coxeter group ( see Proposition 4.1 of Deodhar (1982)). We just state four of them which are based on the definitions given above.Assume (W, S) to be irreducible. Then the following statements are equivalent: 1. W is finite, 2. Ã˜ is finite, 3. The set {((w)|w â‚¬ W} is bounded above,

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