Last update: 26 September 2012
Let and be the elements of the finite Hecke algebra which are determined by
In terms of the basis of these elements have the explicit formulae
where (To define these elements one should adjoin the element to or to The elements and are -analogues of the elements in the group algebra of given by
and the vector spaces and are -analogues of the vector spaces (more precisely, free -modules) of symmetric functions and alternating functions,
respectively, where the action of on is as defined in 1.29.
For let the orbit and the stabilizer of be defined by
Theorem 2.2 below shows that the elements in (2.4) which are indexed by elements of and form bases (over of and This will be a consequence of the following straightening rules. The straightening law for the given in the following Proposition is a generalization of [Mac1995, III §2 Ex. 2].
For let and be as defined in (2.4). Let be a simple root and let be such that Then
Letting if and if then
The first two equalities follow from the definitions of and and the fact that
Let such that Since is in the center of the tiny little affine Hecke algebra generataed by and the
Thus which establishes the third statement.
If then, by definition, If then multiplying the fourth relation in (1.21) by on both the left and the right (and then multiplying by gives
Subtracting the same relation with replaced by gives
Inductively applying this relation yields the result. The first cases are
Proposition 2.1 implies that, for all and
Since form a basis of the elements span By Proposition 2.1, if is on the negative side of a hyperplane that is, if then can be rewritten as a linear combination of such that all terms have on the nonnegative side of By repeatedly applying the relation in Proposition 2.1, can be rewritten as a linear combination of such that all terms have on the nonnegative side of that is, where
If using the fourth relation in (1.21),
where and are some polynomials in such that so that Furthermore unless is in the convex hull of the points in the orbit Thus the coefficient of in is and the coefficient of can be nonzero only if Thus the are linearly independent.
The proof for the cases of and is easier, following directly from (2.5), the fact that is a fundamental chamber for the action of and that if then and in which case (similarly for
For define the Schur function, or Weyl character, by
The straightening law for in (2.5) implies the following straightening law for the Schur functions. If and then, by (2.5) and the definition of
The dot action of the Weyl group on which is appearing here, is ordinary action of on except with the "center" shirted to For the root system of type in Example 1.1, the picture is
The following proposition shows that the Weyl characters are elements of The equaility in part (a) is the Weyl denominator formula, a generalization of the factorization of the Vandermonde determinant In the remainder of this section we shall abuse language and use the term "vector space" in place of "free module".
Let and be as in (1.7) and (2.4) and let be as in (1.8).
Since takes to and permutes the other elements of
Thus the map given by is well defined and it is a bijection since it is invertible.
Since takes to and permutes the other elements of for all and so for all Thus is an element of
If and then
Since is divisible by it follows that is divisible by and thus that is divisible by for all Since the elements are relatively prime in the Laurent polynomial ring and is a unit in is divisible by Since both and are in the quotient is an element of
The monomial appears in with the coefficient 1 and it is the unique term in with Since has highest term with coefficient 1 and is divisible by and it follows that Thus the inverse of the map in (c) is well defined, and is an isomorphism.
Since is a basis of and the map is an isomorphism it follows that is a -basis of
The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).