Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
Last update: 09 April 2012
The elements of index the (isomorphism classes) of simple representations of the quiver.
Consider a sheet of graph paper with diagonals indexed by The content of a box on this sheet of graph paper is
denote a sequence of boxes in a row which has length with the leftmost box of content and the rightmost box of content The set of segments is
The elements of index the (isomorphism classes) of indecomposable (nilpotent) representations of the quiver.
A multisegment is a (unordered) collection of segments, i.e. an elements of
(the numbers in the boxes in the picture are the contents of the boxes).
A multisegment is aperiodic if it does not contain
Pictorially, a multisegment is aperiodic if it does not contain a box of height Let
In types and
The elements of index the isomorphism classes of nilpotent representations of the quiver.
The partial order
Consider an (infinite) sheet of graph paper which has its diagonals labeled consecutively by
The content of a box on this sheet of graph paper is
A segment is a row of boxes on a sheet of graph paper with diagonals indexed by
Consider a graph paper with diagonals indexed by A segment is a sequence of boxes
in a row with the leftmost box of content and the rightmost box of content A multisegment is a (unordered) collection of segments. For example
for all other segments (the numbers in the boxes in the picture are the contents of the boxes). Alternatively a multisegment can be viewed as a function
The set of segments is ordered by inclusion. Define
and so the multisegment can be specified by the numbers
Define a partial order on multisegments by
are segments define a degeneration
is elementary if
Pictorially a degeneration takes
Let be the quiver with
Fix an graded vector space
provides a bijection
Let and be multisegments and let and be the corresponding orbits in Then the following are equivalent:
for some sequence of elementary degenerations
for all segments Find (THIS STILL NEEDS DOING) a sequence of elementary degenerations which takes to i.e.
Hecke algebra representations
Let be the affine Hecke algebra at an root of unity so that (all if desired). For each
where and The simple modules are indexed by
and are determined by the equations
in the Grothendieck group of modules.
The Fock space representation of
The crystal graph
be a multisegment and assume that it is ordered so that
These conditions are equivalent to saying that
The weight is integrally dominant,
where is integrally dominant and is longest in its coset
Then, ignoring 0s, read the sequence of +1s, -1s left to right and successively cancel adjacent pairs to get a sequence of the form
The -1s in this sequence are the normal nodes and the +1s are the conormal nodes. The good node is the leftmost normal node and the cogood node is the rightmost conormal node.
If this algorithm is being executed where then take
when and and
In type is the connected component of in the crystal graph
is the crystal graph of
and identify with the partition which has boxes in row Let
and define an imbedding
where the entries are the entries of read in Arabic reading order.
The tensor product representation
The dimensional simple module of highest weight is given by
+1 over each in
-1 over each in
0 over each
Then the action on is given by
where the first sum is over all which are obtained from by changing a to and the second sum is over all which are obtained from by changing a to
The Fock space
Let for Define
Define an action of on by
These formulas make into a module.
so that the multisegments form a
basis of then
The permutations of the sequence
are indexed by the elements of
where is the number of nodes after pairing. The group
acts on the pairs by changing a pair to For each define
The first statement is clear. To obtain the second statement
A Schur-Weyl duality connection to affine Hecke algebras
A multisegment is a collection of rows of boxes (segments) placed on graph paper. We can label this multisegment by a pair of weights
(the numbers in the boxes in the picture are the contents of the boxes). The construction forces the condition
and since we want to consider unordered collections of boxes it is natural to take the following pseudo-lexicographic ordering on the segments,
when we denote the multisegment by a pair of weights In terms of weights the conditions (a), (b) and (c) can be restated as (note that in this case both and are integral)
is a weight of where is the number of boxes in
is integrally dominant,
with integrally dominant and maximal length in the coset
Let be a multisegment with boxes and number the boxes of from left to right (like a book). Define
so that is the "parabolic" subalgebra of corresponding to the multisegment Define a one-dimensional module
for and such that is not at the end of its row.
Let be of type and let be the functor
where The standard module for the affine Hecke algebra is
as defined in (4.1). It follows from the above discussion that these modules are naturally indexed by multisegments The following proposition shows that this standard module coincides with the usual standard module for the affine Hecke algebra as considered by Zelevinsky [Ze2] (see also [Ar], [CG] and [KL]).
Let be a multisegment determined by a pair of weights with integrally dominant. Let be the one dimensional representation of the parabolic subalgebra of the affine Hecke algebra defined in (???). Then
To remove the constants that come from the difference between and the affine braid group action in Theorem 6.17a should be normalized so that
By Proposition 4.3a,
as a vector space. Let
be the standard basis of
If we let the symmetric group act on by permuting the tensor factors then
is the parabolic subgroup of which stabilizes the vector
This shows that, as vector spaces,
For notational purposes let
and let be the image of in
Since is integrally dominant and has weight it must be a highest weight vector. We will show that acts on by the constant where is the content of the box of the multisegment (read left to right and top to bottom like a book).
Consider the projections
and acts as the identity on the last factors of
and for each (the first components of)
form a highest weight vector of weight in
It is the "highest" highest weight vector of
with respect to the ordering in Lemma 4.2 and thus it is deepest in the filtration constructed there. Note that the quantum Casimir element acts on the space in (6.29) as the constant
times a unipotent transformation, and the unipotent transformation must preserve the filtration coming from Lemma 4.2. Since
is the highest weight vector of the smallest submodule of this filtration (which is isomorphic to a Verma module by Lemma 4.2b) it is an eigenvector for the action of the quantum Casimir. Thus, by (2.11) and (2.13), acts on
by the constant
(see [LR]). Since commutes with for this also specifies the action of on
The explicit matrix
for this case ( of type and ) is well known (see, for example, the proof of [LR, Prop. 4.4]) and given by
Since acts by on the and tensor factors of and commutes with the projection it follows that
if is not a box at the end of a row of This analysis of the action of on shows that there is an homomorphism
This map is surjective since is generated by (the action on generates all of
Finally, (6.28) guarantees that it is an isomorphism.
In the same way that each weight has a normal form
every multisegment has a normal form
The element in the normal form
of can be constructed combinatorially by the following scheme. We number (order) the boxes of in two different ways.
First ordering: To each box of associate the following triple
where, if a box is the leftmost box in a row "the box to its left" is the rightmost box in the same row. The lexicographic ordering on these triples induces an ordering on the boxes of
Second ordering: To each box of associate the following pair
The lexicographic ordering of these pairs induces a second ordering on the boxes of
If is the permutation defined by these two numberings of the boxes then For example, for the multisegment displayed in (6.24) the numberings of the boxes are given by
and the normal form of is
Let be of type and and let
as defined in (4.1). It is known (a consequence of Proposition 6.27 and Proposition 4.3c) that is always a simple module or 0. Furthermore, all simple modules are obtained by this construction. See [Su] for proofs of these statements. The following theorem is a reformulation of Proposition 4.12 in terms of the combinatorics of our present setting.
Let and be multisegments with boxes (with and assumed to be integral) and let
be their normal forms. Then the multiplicities of in a Jantzen filtration of are given by
where is the Kazhdan-Lusztig polynomial for the symmetric group
Theorem 6.31 says that every decomposition number for affine Hecke algebra representations is a Kazhdan-Lusztig polynomial. The following is a converse statement which says that every Kazhdan-Lusztig polynomial for the symmetric group is a decomposition number for affine Hecke algebra representations. This statement is interesting in that Polo [Po] has shown that every polynomial in
is a Kazhdan-Lusztig polynomial for some choice of and permutations Thus, the following proposition also shows that every polynomial arises as a generalized decomposition number for an appropriate pair of affine Hecke algebra modules.
Then, for each pair of permutations the Kazhdan-Lusztig polynomial for the symmetric group is equal to
Since and are both regular,
and the standard and irreducible modules
ranging over all Thus, this statement is a corollary of Proposition 4.12.
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