Last update: 27 February 2013
In this section we review the generalization of standard Young tableaux in [Ram0401326] which is used to construct representations of the affine Hecke algebras Then we show how this theory can be transported to provide combinatorial constructions of simple modules for the cyclotomic Hecke algebras and This approach shows that Jucys-Murphy type elements in the cyclotomic Hecke algebras arise naturally as images of the elements in the affine Hecke algebra. The standard Jucys-Murphy type theorems then follow almost immediately from standard affine Hecke algebra facts.
A partition is a collection of boxes in a corner. We shall conform to the conventions in [Mac1995] and assume that gravity goes up and to the left.
Any partition can be identified with the sequence where is the number of boxes in row of The rows and columns are numbered in the same way as for matrices. In the example above we have If and are partitions such that for all we write The skew shape consists of all boxes of which are not in Any skew shape is a union of connected components. Number the boxes of each skew shape along major diagonals from southwest to northeast and
Let be a skew shape with boxes. A standard tableau of shape is a filling of the boxes in the skew shape with the numbers such that the numbers increase from left to right in each row and from top to bottom down each column.
Let If is a positive real number then the function
is a bijection. The elements of index the in
A placed skew shape is a pair consisting of a skew shape and a content function
This is a generalization of the usual notion of the content of a box in a partition (see [Mac1995] I §1 Ex. 3).
Suppose that is a placed skew shape such that takes values in One can visualize by placing on a piece of infinite graph paper where the diagonals of the graph paper are indexed consecutively (with elements of from southeast to northwest. The content of a box is the index of the diagonal that is on. In the general case, when takes values in one imagines a book where each page is a sheet of infinite graph paper with the diagonals indexed consecutively (with elements of from southeast to northwest. The pages are numbered by values and there is a skew shape placed on page The skew shape is a union of the disjoint skew shapes on each page,
and the content function is given by
for a box
The following diagrams illustrate standard tableaux and the numbering of boxes in a skew shape
The following picture shows the contents of the boxes in the placed skew shape such that the sequence
The following picture shows the contents of the boxes in the placed skew shape such that
This “book” has two pages, with page numbers 0 and
A finite dimensional is calibrated if it has a basis such that for each and each in the basis
This is the class of representations of the affine Hecke algebra for which there is a good theory of Young tableaux [Ram2003].
The following theorem classifies and constructs all irreducible calibrated representations of the affine Hecke algebra The construction is a direct generalization of A. Young’s classical “seminormal construction” of the irreducible representations of the symmetric group [You1931]. Young’s construction was generalized to Iwahori-Hecke algebras of type A by Hoefsmit [Hoe1974] and Wenzl [Wen1988] independently, to Iwahori-Hecke algebras of types B and D by Hoefsmit [Hoe1974] and to cyclotomic Hecke algebras by Ariki and Koike [AKo1994]. In (3.7) and (3.11) below we show how all of these earlier generalizations of Young’s construction can be obtained from Theorem 3.8. Some parts of Theorem 3.8 are originally due to I. Cherednik, and are stated in [Che1987, §3].
Theorem 3.8. ([Ram1997, Theorem 4.1]) Let be a placed skew shape with n boxes. Define an action of on the vector space
by the formulas
where is the same as except that the entries and are interchanged,
and denotes the box of containing the entry
Remark 3.9. All of the irreducible modules for the affine Hecke algebra have been classified and constructed by Kazhdan and Lusztig [KLu0862716]. The construction in [KLu0862716] is geometric and noncombinatorial. It is nontrivial (but not very difficult) to relate the construction of Theorem 3.8 and the classification in [KLu0862716].
A finite dimensional module is calibrated if it has a basis such that for each and each in the basis
Let us show how Theorem 3.8, Theorem 2.2 and Theorem A.13 provide explicit constructions of simple calibrated The resulting construction is a generalization of the construction of given by Ariki [Ari1995] (as amplified and applied in [HRa1998]). Comparing the following machinations with those in [HRa1998, §3] (where more pictures are given) will be helpful.
The on induces an action of on the simple as in (A.1) of the Appendix, and this action takes calibrated modules to calibrated modules since the on preserves the subalgebra The action on simple calibrated modules can be described combinatorially as follows.
If is a placed skew shape with boxes and define
and denotes the content function defined by for all boxes To make this definition we are identifying the set with One can imagine the placed skew shape as a book with pages numbered by values and a skew shape on each page. The action of
If is a standard tableau of shape let denote the same filling of as but viewed as a standard tableaux of shape
Let be the except twisted by the automorphism see (A.1) in the Appendix. It follows from the formulas in Theorem 3.8 that the map
is an isomorphism. Indeed, since and
where denotes the on as in (A.1). Identify with via the isomorphism in (3.12).
Let be a placed skew shape with boxes and let be the stabilizer of under the action of The cyclic group is a realization of the inertia group of and
where is the smallest integer between 1 and such that and is the order of The elements of are all isomorphisms. Since is a cyclic group the irreducible are all one-dimensional and the characters of these modules are given explicitly by
since is a primitive root of unity. The element
is the minimal idempotent of the group algebra corresponding to the irreducible character It follows (from a standard double centralizer result, [Bou1958]) that, as an
and is the irreducible with character The following theorem now follows from Theorem A.13 of the Appendix.
Theorem 3.15 Let be a placed skew shape with boxes and let be the simple calibrated constructed in Theorem 3.8. Let be the inertia group of corresponding to the action of on defined by (A.1). If is the minimal idempotent of given by (3.13) then
is a simple calibrated
Theorem 3.15 provides a generalization of the construction of the which was given by Ariki [Ari1995] and extended and applied in [HRa1998].
Many (usually all) of the simple can be constructed with Theorems 2.8 and 3.8.
If is a skew shape define
so that is the set of boxes such that there is no box of immediately above or immediately to the left of
Theorem 3.17. Fix and let be a placed skew shape with boxes. If
then the is a simple (via Theorem 2.8).
Let be a placed skew shape with boxes and let be the simple of Theorem 3.8. Let be the representation corresponding to By the formulas in Theorem 3.8 the matrix is diagonal with eigenvalues for where is the set of standard tableaux of shape
The boxes are exactly the northwest corner boxes of and so the minimal polynomial of is
Thus the condition is exactly what is needed for to be an via Theorem 2.8.
Theorem 3.18. If the cyclotomic Hecke algebra is semisimple, then its simple modules are the modules constructed in Theorem 3.17, where is an of partitions with a total of boxes and is the content function determined by
The equations for determine the values for all boxes and thus the pair defines a placed skew shape. By a Theorem of Ariki [Ari1994], is semisimple if and only if
where and These conditions guarantee that is a placed skew shape and that the defined in Theorem 3.8 is well defined and irreducible. The reduction in Theorem 2.8 makes into a simple and a count of standard tableaux (using binomial coefficients and the classical identity for the symmetric group case) shows that
where the sum is over all with a total of boxes. Thus the are a complete set of simple
Theorem 3.18 demonstrates that the construction of simple modules for by Ariki and Koike [AKo1994, Theorem 3.7], for by Hoefsmit [Hoe1974], for by Hoefsmit [Hoe1974] and Wenzl [Wen1988] (independently), and for and by Young [You1929,You1931], are all special cases of Theorem 3.8.
The following result is well known, but we give a new proof which shows that the cyclotomic Hecke algebra analogues of the Jucys-Murphy elements which have appeared in the literature (see [BMM1993], [Ram1997], [DJM1995] and the references there) come naturally from the affine Hecke algebra
Corollary 3.20. Let and be the cyclotomic Hecke algebras defined in (1.1) and (1.2).
(a) The elements generate the subalgebra Inductive use of the relation (1.4) shows that where is the homomorphism in Proposition 2.5. Thus the subalgebra of generated by the element is the image of under the homomorphism
(b) is an immediate consequence of Theorem 3.18, the construction described in Theorem 3.8, and the fact that
(c)The elements and generate the subalgebra and the images of these elements under the homomorphism are the elements and
The proof of part (d) uses the construction described in Theorem 3.15 and is analogous to the proof of part (b).
It is an immediate consequence of (1.12) and the proof of Corollary 3.20 that
where The following proposition shows that this inclusion is an equality.
Proposition 3.22. If the cyclotomic Hecke algebra is semisimple then its center
where and is the ring of symmetric polynomials in
By (1.12) Thus, since where is the surjective homomorphism of Proposition 2.5, it follows that
For the reverse inclusion we need to show that the action of the elements distinguishes the simple
Let be an of partitions with a total of boxes and let be the corresponding simple as constructed by Theorem 3.18 (and Theorem 3.8). If is a standard tableau of shape then act on by the multiset of values The elementary symmetric functions act on by the values Note that does not depend on the choice of the standard tableau (since We show that the simple module is determined by the values This shows that the simple modules are distinguished by the elements of
Let us explain how the values determine the placed skew shape There is a unique (unordered) multiset of values such that for all The are determined by the equation
From the previous paragraph, the multiset must be the same as the multiset In this way the values determine the multiset
The multiset is a disjoint union of multisets such that each is in a single of (see (3.4)). These cosets are determined by the values and thus there is a one-to-one correspondence between the and the multisets Then the nonempty diagonals of are determined by the values in and the lengths of the diagonals of are the multiplicities of the values of the elements of This information completely determines for each Thus is determined by the values
Remark 3.23. In the language of affine Hecke algebra representations (see [Ram2003]) the proof of Proposition 3.22 shows that (when is semisimple) the simple all have different central characters (as modules).
Remark 3.24. The elements and from Corollary 3.20 cannot be used to obtain a direct analogue of Proposition 3.22 for This is because all of the appearing in the decomposition (3.14) will have the same central character. However, when is odd, the natural analogue of Proposition 3.22 does hold for Iwahori-Hecke algebras of type In that case, every simple has trivial inertia group and the decomposition in (3.14) has only one summand.
This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.
Research partially supported by the Natioanl Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).