Last update: 14 November 2012
This section describes the sets in the case of the root system of Section 5.2 when a primitive root of unity,
Let Identify with a sequence
For the purposes of representation theory (see Theorem 3.6) indexes a central character (see Section 2.3) and so can safely be replaced by any element of its In this case is the symmetric group, acting by permuting the sequence
The cyclic group of order generated by acts on F ix a choice of a set of coset representatives of the cosets in Replace with the sequence obtained by rearranging its entries to group entries in the same so that
where are distinct representatives of the cosets in and each is a sequence of the form
As in Section 5.11 this decomposition of into groups induces decompositions
and it is sufficient to analyze the case when consists of only one group, i.e., all the entries of t are in the same coset.
Now assume that
Consider a page of graph paper with diagonals labeled by from southwest to northeast. For each local region we will construct an configuration of boxes for which the standard tableaux defined below will be in bijection with the elements of For each the configuration will have a box numbered on each diagonal which is labeled There are an infinite number of such diagonals containing a box numbered since the diagonals are labeled in an fashion, but each strip of consecutive diagonals labeled will contain boxes. The content of a box (see [Mac1995, I, Section 1, Exercise 3]) is
We will use to organize the relative positions of the boxes in adjacent diagonals:
Thus, determines the number of boxes in each diagonal and determines the relative positions of the boxes in adjacent diagonals. This information completely determines the configuration of boxes associated to the pair
A standard tableau is an filling of the boxes with such that
where denotes the entry in An standard tableau corresponds to a permutation in via the correspondence
Example. Suppose and
then the corresponding configuration of boxes and a sample standard tableau are
This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.
Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).