Last update: 13 November 2012
In this section we shall show that the combinatorics of local regions is a generalization of the combinatorics of standard Young tableaux. Let us first make some general definitions, which we will show later provide generalizations of standard objects in the Young tableaux theory. This section is a (purely combinatorial) study of the local regions in the form which appears in (4.3), and therefore corresponds to the representation theory of affine Hecke algebras when is not a root of unity.
Let be dominant and let
as in (4.3).
Remarks. (1) Theorem 4.6(b) shows that, up to a shift, the set has a unique maximal and a unique minimal element and is an interval in the weak Bruhat order. This is the fundamental importance of the notions of the row reading and the column reading tableaux. Theorem 6.9 in Section 6 will show how Theorem 4.6(b) is a generalization of a Young tableaux result of Björner and Wachs [Bjo1988, Theorem 7.2].
(2) The definition of skew local regions is forced by the representation theory of the affine Hecke algebra (see Theorem 3.6, the classification of irreducible calibrated representations). In Proposition 6.4 below we shall show that the skew local regions and the ribbons are generalizations of the skew shapes and border strips which are used in the theory of symmetric functions [Mac1995, I, Section 5 and I, Section 3, Exercise 11]
(3) The axial distances control the denominators which appear in the construction of irreducible representations of the affine Hecke algebra in Theorem 3.5. In Section 6.1 we shall see how they are analogues of the axial distances used by A. Young [You1931] in his constructions of the irreducible representations of the symmetric group.
To summarize, a brief dictionary between local regions combinatorics and the Young tableaux combinatorics:
The remainder of this section and the next section explain in greater detail the conversions indicated in this dictionary.
Let be an orthonormal basis of so that each sequence is identified with the vector The root system of type is given by the sets
The Weyl group is the symmetric group, acting by permutations of the
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. We shall always use the word diagonal to mean a major diagonal. In the example above and the diagonals of (from southwest to northeast) contain 1, 1, 1, 2, 3, 3, 2, 2, 2, and 1 box, respectively.
If and are partitions such that for all write The skew shape consists of all boxes of which are not in Let be a skew shape with boxes. Number the boxes of each skew shape along diagonals from southwest to northeast and
See Example 5.8 below. 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 be the set of standard tableaux of shape Given a standard tableau of shape define the word of to be the permutation
where is the entry in of the standard tableau.
Let be a skew shape with boxes. Imagine placing on a piece of infinite graph paper where the diagonals of the graph paper are indexed consecutively (with elements of from southwest to northeast.
The content of a box is
Identify the sequence
The pair is a placed skew shape. It follows from the definitions in Section 5.1 that
where northwest means strictly north and weakly west.
Examples 5.8. The following diagrams illustrate standard tableaux and the numbering of boxes in a skew shape
The word of the standard tableau is the permutation (in one-line notation).
The following picture shows the contents of the boxes in the placed skew shape with
In this case
Theorem 5.9. Let be a placed skew shape and let be as defined in (5.7). Let be the set of standard tableaux of shape and let be the set defined in Section 5.1. Then the map
where is as defined in (5.4), is a bijection.
If is a permutation in then
The theorem is a consequence of the following chain of equivalences:
The filling is a standard tableau if and only if, for all
These conditions hold if and only if
which hold if and only if
Finally, these are equivalent to the conditions and
We have described how one can identify placed skew shapes with certain pairs One can extend this conversion to associate placed configurations of boxes to more general pairs The resulting configurations are not always skew shapes.
Let be a pair such that is a dominant integral weight and (The sequence is a dominant integral weight if and for all If satisfies the condition
then will determine a placed configuration of boxes (see Theorem 4.6). As in the placed skew shape case, think of the boxes as being placed on graph paper where the boxes on a given diagonal all have the same content. (The boxes on each diagonal are allowed to slide along the diagonal as long as they do not pass through the corner of a box on an adjacent diagonal.) The sequence describes how many boxes are on each diagonal and the set determines how the boxes on adjacent diagonals are placed relative to each other. We want
where the boxes are numbered along diagonals in the same way as for skew shapes, southeast means weakly south and strictly east, and northwest means strictly north and weakly west.
If we view the pair as a placed configuration of boxes then the standard tableaux are fillings of the boxes in the configuration with such that, for all
As in (5.6) the permutation in which corresponds to the standard tableau is The following example illustrates the conversion.
Example. Suppose and
The placed configuration of boxes corresponding to is as given below.
The general case, when is an arbitrary element of and is handled as follows. First group the entries of according to their in Each group of entries in can be arranged to form a sequence
Fix some ordering of these groups and let
be the rearrangement of the sequence with the groups listed in order. Since and are in the same orbit it is sufficient to analyze corresponds to the central character of the corresponding affine Hecke algebra representations and thus any convenient element of the orbit is appropriate, see Section 2.3).
The decomposition of into groups induces decompositions
where Each pair is a placed shape of the type considered in the previous subsection and we may identify with the book of placed shapes We think of this as a book with pages numbered by the values and with the placed configuration determined by on page In this form the standard tableaux of shape are fillings of the boxes in the book with the numbers such that the filling on each page satisfies the conditions for a standard tableau in Section 5.10.
Example. If then one possibility for is
In this case
If where and
then the book of shapes is
where the numbers in the boxes are the contents of the boxes. The filling
is a standard tableau of shape This filling corresponds to the permutation
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).