Last update: 8 June 2014
This is a typed copy of Peter Norbert Hoefsmit's thesis Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type published in August, 1974.
A partition of is an ordered set of positive numbers such that arbitrary. Such a partition is said to have parts and its length, is Thus is a partition of It may also be written and a similar notation will be used elsewhere.
We represent by a Young diagram, having squares in the first row, squares in the second row and so on, the squares of the rows making a column. is said to have shape Thus is the Young diagram of shape The square in the row and column of is said to have coordinates and is called the
Let denote the number of squares appearing in the column of The Young diagram obtained by interchanging the rows and columns of is called the conjugate of and the partition is called the conjugate of
The letters may be arranged in the squares of in ways. Each such arrangement is called a tableau of shape A tableau is called a standard tableau if the letters in every row increase from left to right and in every column from top to bottom. Thus the tableaux (ii) and (iii) are standard while (i) is not. The tableau of shape with the letters arranged in consecutive order in the rows, starting with the first square in the first row is called the canonical tableau of shape The tableau (iii) above is the canonical tableau of shape
The number, of standard tableaux of shape is determined as follows (see [Rob1961], p. 44). The of determines the consisting of the along with the squares to the right in the row and the squares below in the column. Thus the length of the is Then
A double partition of is an ordered pair of partitions with If and we write the double partition as where and where We allow either or to be a partition of in the above, i.e. let denote the empty partition. For a partition of and are distinct double partitions of We represent by an ordered pair of Young diagrams called the Young diagram of shape is considered to have rows, where the row of is the row of and the row of is the row of The Young diagrams of shape and are taken to be and The squares of are identified by their coordinates in the diagrams and Thus the square of which is in the row and in the column of (resp. is called the of (resp. and has coordinates Hence distinct squares of can have the same coordinates (for instance, the first square in the first row of and the first square in the row of both have coordinates where is as above).
A tableau of shape
is any arrangement of the letters in
Thus a tableau is an ordered pair
where, for complementary subsets and of
denotes any arrangement of the letters of in
and denotes any arrangement of the letters of in
is a standard tableau if the arrangement of the letters is in increasing order in the rows and columns of both
and Thus, Figure 2
is a standard tableau of shape
The canonical tableau of shape is the tableau where the letters are arranged
consecutively in the rows of starting with the first square in the
first row of The number,
of standard tableaux of shape is
We order the standard tableaux of a given shape as follows:
Definition (1.1.4) Let denote standard tableaux of shape We say precedes if the letters appear in the same row in both tableaux but the letter appears in a lower row in than in The enumeration of the standard tableaux according to their ordering is called the last letter sequence.
Thus in the last letter sequence all tableaux which have the letter in the last row precede those which have in the second to the last row. These latter tableaux precede those which have in the third to the last row and so on. Those tableaux which have in the same row are arranged by the same scheme according to the position of the letter and so on. It is evident that the canonical tableau is the first tableau in this ordering.
We give an example of this ordering. For the double partition there are 20 standard tableaux. Arranged according to the last letter sequence they are, Finally we define the notion of axial distance.
Definition (1.1.5) For squares and in a Young diagram with coordinates and respectively, define the axial distance, from to to be
Axial distance has a simple graphical interpretation. Suppose the squares and are in the same diagram of Starting from proceed by any rectangular route one square at a time until is reached. Counting for each step made upwards or to the right and for each step made downwards or to the left, the resultant number of steps made is the axial distance from to . For squares belonging to distinct diagrams, axial distance is the distance of any rectangular route, counted as above, in the diagram obtained by superimposing upon
Definition (1.1.16) The axial distance from the letter to the letter in a tableau is the axial distance from the square of in which appears to the square of in which appears.
Thus in Figure 2 the axial distance from to is the axial distance from to is the axial distance from to is and the axial distance from to is
We briefly describe the irreducible semi-normal representations of the symmetric group on letters. The conjugacy classes of are parameterized by the partitions of In ([You1930]) Young constructed for each partition of an irreducible representation of of degree by constructing primitive idempotents, the natural idempotents, in the group algebra from the standard tableaux of shape The distinctive feature of these representations is that they are integral, i.e., matrix representations afforded by the minimal left ideals generated by these idempotents have entries in In a subsequent paper ([You1931]) Young constructed an equivalent form of these representations by means of the semi-normal idempotents. While the corresponding matrix representations are not integral, Young showed an elegant construction to hold for the matrices of the transpositions by means of the standard tableaux. For a tableau of shape let denote the tableau obtained by interchanging the letters and in If is a standard tableau and the letters and do not occur either in the same row or column of then is again a standard tableau. Young's fundamental theorem giving the semi-normal form of the representations of can now be stated as follows.
Theorem (1.2.1) Let be the arrangement of the standard tableaux of shape according to the last letter sequence. To construct the matrix representing in the irreducible representation of corresponding to place
|(i)||in the entry where has and in the same row,|
|(ii)||in the entry where has and in the same column,|
|(iii)||the matrix in the and entries where and is the axial distance (see (1.1.6)) from to in|
The importance of the semi-normal form is that it provides an inductive construction of the irreducible representations of and moreover is defined in terms of the generators and relations of
Young's fundamental theorem can be extended to yield representations of corresponding to double partitions of If is a partition of and is a partition of with let denote the representation of induced from the direct product representation of the subgroup of The representation is called the outer product representation of and For a standard tableau of shape if the letters and do not occur either in the same row or column of or is again a standard tableau. Arranging the standard tableaux of shape according to the last letter sequence, we have (see [Rob1961], p. 54).
Theorem (1.2.2) To construct the matrices presenting in the outer product representation of apply the construction given in Theorem (1.2.1) to the standard tableau of shape setting in (iii) if the letters and belong to distinct tableaux
The hyperoctahedral group of order is the group of signed permutations on letters. It can be regarded as acting on an orthonormal basis of by means of permutations and sign changes. Denote the sign change, by The set of transpositions and the first sign change, generate In ([You1929]) Young showed the conjugacy classes of to be parameterized by double partitions of and constructed for each double partition an irreducible representation of of degree by constructing primitive idempotents in analogous to the natural idempotent of Young did not construct the analogous of the semi-normal idempotents for It is implicit in his work, however, that a semi-normal form can be constructed for the representations using Theorem (1.2.2). In particular, using the matrices for the transpositions in the outer product representation we need only construct a matrix for the first sign change which satisfies the relations of It can readily be shown (see corollary (2.2.15)), using the inductive ordering provided by the last letter sequence on the standard tableaux of shape that
Theorem (1.2.3) To construct the matrices representing in the irreducible representations of apply Theorem (1.2.2). To construct the matrix representing place
|1.||in the entry if the letter appears in the tableau of|
|2.||in the entry if the letter appears in the tableau of|
Thus, analogous to Theorem (1.2.1), the irreducible representations of can be defined inductively and in terms of generators and relations.