Last update: 13 November 2012
In this section we shall explain how the definitions in Section 5.1 correspond to classical notions in Young tableaux theory. As in the previous section let be the root system of Type as given in Section 5.2. For clarity, we shall state all of the results in this section for placed shapes such that is dominant and integral, i.e., with and This assumption is purely for notational clarity.
Let be a local region such that is dominant and integral. Let and let be the corresponding standard tableau as defined by the map in Theorem 5.9. Then it follows from the definitions of and in (5.6) and (5.4) that
where is the box of containing the entry
In classical standard tableau theory the axial distance between two boxes in a standard tableau is defined as follows. Let be a partition and let be a standard tableau of shape Let and let and be the boxes which are filled with and respectively. Let and be the positions of these boxes, where the rows and columns of are numbered in the same way as for matrices. Then the axial distance from to in is
(see [Wen1988]). Rewriting this in terms of the local region determined by (5.7),
where is the permutation corresponding to the standard tableau and is the distance defined in (f) of Section 5.1. This shows that the axial distance defined in (f) of Section 5.1 is a generalization of the classical notion of axial distance. These numbers are crucial to the classical construction of the seminormal representations of the symmetric group given by Young (see Remark (3) of Section 5.1).
The following proposition shows that, in the case of a root system of type A, the definition of skew local region coincides with the classical notion of a skew shape.
Proposition 6.4. Let be a local region with dominant and integral. Then the configuration of boxes associated to is a placed skew shape if and only if is a skew local region.
We shall show that if the placed configuration corresponding to the pair has any blocks of the forms
then there exists a such that violates one of the two conditions in (c) of Section 5.1. This will show that if is a skew local region then the corresponding placed configuration of boxes must be a placed skew shape. In the pictures above the shaded regions indicate the absence of a box and, for reference, we have labeled the boxes with
Case (1). Create a standard tableau such that the block is filled with
by filling the region of the configuration strictly north and weakly west of box c in row reading order (sequentially left to right across the rows starting at the top), putting the next entry in box c, and filling the remainder of the configuration in column reading order (sequentially down the columns beginning at the leftmost available column). Let be the permutation in which corresponds to the standard tableau Let denote the box containing in Then, using the identity (6.2),
since the boxes and are on the same diagonal. However,
and so condition (c)(1) of Section 5.1 is violated.
Case (2). Create a standard tableau such that the block is filled with
by filling the region weakly north and strictly west of box c in column reading order, putting the next entry in box c, and filling the remainder of the configuration in row reading order. Using this standard tableau the remainder of the argument is the same as for Case (1).
Case (3). Create a standard tableau such that the block is filled with
by filling the region strictly north and strictly west of box b in column reading order, putting the next entry in box b, and filling the remainder of the configuration in row reading order. Let be the permutation in corresponding to and let denote the box containing in Then
since and are on the same diagonal. Hence, condition (c)(1) of Section 5.1 is violated.
Let and describe a placed skew shape (a skew shape placed on infinite graph paper). Let be the corresponding local region as defined in (5.7). We will show that every is calibratable for every
Let and let be the corresponding standard tableau of shape Consider a block of boxes of If these boxes are filled with
then either or In both cases we have and it follows that and are not on the same diagonal. Thus
and so satisfies condition (a) in the definition of calibratable.
The same argument shows that one can never get a standard tableau in which and occur in adjacent boxes of the same diagonal and thus it follows that satisfies condition (b) in the definition of calibratable. Thus is a skew local region.
Classically, a border strip (or ribbon) is a skew shape which contains at most one box in each diagonal. Although the convention, [Mac1995, I, Section 1 p. 5], is to assume that border strips are connected skew shapes we shall not assume this.
Recall from (b) of Section 5.1 that a placed shape is a placed ribbon shape if is regular, i.e., for all
Proposition 6.6. Let be a placed ribbon shape such that is dominant and integral. Then the configuration of boxes corresponding to is a placed border strip.
Let be a placed ribbon shape with dominant and regular. Since is regular, for all In terms of the placed configuration is the diagonal that is on. Thus the configuration of boxes corresponding to contains at most one box in each diagonal.
Example. If and then the placed configuration of boxes corresponding to is the placed border strip
where the boxes are labeled with their contents.
Let be a placed shape with dominant and integral (i.e., with and and view as a placed configuration of boxes. In terms of placed configurations, conjugation of shapes is equivalent to transposing the placed configuration across the diagonal of boxes of content 0. The following example illustrates this.
Example. Suppose and Then the placed configuration of boxes corresponding to is
in which the shaded box is not a box in the configuration.
The minimal length representative of the coset is the permutation
We have and
Thus the configuration of boxes corresponding to the placed shape is
Let be a placed shape such that is dominant and integral and consider the placed configuration of boxes corresponding to The minimal box of the configuration is the box such that
There is at most one box in each diagonal satisfying Thus, guarantees that the minimal box is unique. It is clear that the minimal box of the configuration always exists.
The column reading tableaux of shape is the filling which is created inductively by
The row reading tableau of shape is the standard tableau whose conjugate is the column reading tableaux for the shape (the conjugate shape to
Recall the definitions of the weak Bruhat order and closed subsets of roots given after Eq. (4.5).
Theorem 6.9. Let be a placed shape such that is dominant and integral (i.e., with and Let and be the column reading and row reading tableaux of shape respectively, and let and be the corresponding permutations in Then
where denotes the complement of in and
(a) Consider the configuration of boxes corresponding to If then either or is in the same diagonal and southeast of Thus when we create we have that
where the northwest is in a very strong sense: There is a sequence of boxes
such that is either directly above or in the same diagonal and directly northwest of In other words,
So, from the formula for in (5.4) we get
where is the permutation in which corresponds to the filling and is the closure of in It follows that
(b) There are at least two ways to prove that One can mimic the proof of part (a) by defining the maximal box of a configuration and a corresponding filling. Alternatively one can use the definition of conjugation and the fact that The permutation is the unique minimal element of and the conjugate of is the unique minimal element of We shall leave the details to the reader.
(c) An element is an element of if and only if and Thus consists of those permutations such that
Since the weak Bruhat order is the ordering determined by inclusions of it follows that is the interval between and
Example. Suppose and The minimal and maximal elements in are the permutations
The permutations correspond to the standard tableaux
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).