Last update: 20 August 2013
Proposition 2.6 If is defined in terms of diagrams, with then generate
We will examine the permutation
Definition 2.7 A reduced word for is an expression of minimal length. The number is the length of often denoted
Recall (or note) that the length of a permutation is equal to the number of crossings in the diagram for (with suitable conventions on the diagram – no superfluous crossings, e.g.). The length is also equal to the number of inversions of the permutation. We will also need in what follows the fact that every presentation of a permutation of a product of has its number of generators in the presentation congruent to modulo two.
Definitions 2.8 The set of inversions of is the set The sign is
Note that in each permutation has the same cycle type as its inverse, and therefore for any character one has By an abuse of notation, we will sometimes denote by
For the trivial representation of we have The associated character is One has that
Now, let us consider the alternating representation of Denoted by it is generated by sending each to the matrix One has (Again, note that is independent of the choice of presentation of This representation gives rise to the character Since characters are class functions, we only need consider their behavior on cycle structures; we have that since
Recall that the minimal central idempotent for the trivial representation was For the alternating representation, we have
Homework Problem 2.9 Find analogs of the above representations, characters, and idempotents for the Hecke algebras.
We have seen the two one-dimensional representations of Now we will consider the permutation representation, wherein sends each permutation to the appropriate permutation matrix. This representation (unlike the trivial and alternating representations) is not irreducible; in fact Note that the determinant of the permutation matrix is so by taking the composition of the permutation representation and the determinant map one (essentially) obtains the alternating representation.
Now we consider the representation of on words of length Let be letters (i.e. indeterminate symbols), and let acts on from the right by
Definition 2.10 Let be commuting variables. Define the weight of to be
Note, then, that upon re-ordering, where is the number of occurrences of the letter in the word Note also that for
Definition 2.11 A sequence such that and is called a composition of denoted
Definition 2.12 Fix a composition The following are all submodules of
The fact that the above are submodules follows from the stability of the weight operator under the action of
Definition 2.13 The weighted trace of a permutation denoted is
We would like to compute the characters of the representations Note that Now, since the trace tr of a permutation is completely determined by the trace of the associated cycle structure, the same is true of the weighted trace wtr. Thus to compute the traces it suffices to compute the weighted traces
Lemma 2.14 If so then
The above works since only acts on the first many characters, only acts on the through characters, etc..
Proposition 2.15 If and then
Recall that Thus
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.