Representations of Hecke Algebras of Finite Groups with BN-Pairs of Classical Type
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
Last update: 8 June 2014
Notes and References
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.
Chapter 2.Representations of the Generic Ring Corresponding to a Coxeter System of Classical Type
Partitions and Tableaux
Let be a subgroup of a finite group and let be a field of characteristic zero. Set
in the group algebra Then affords the
of and the left
affords the induced representation
Definition (2.1.1)
The Hecke algebra is the subalgebra of
given by
The Hecke algebra acts on by right multiplication and the action defines an isomorphism between
and the endomorphism algebra
The double coset sums
form a basis for
(see [Ste1967], Lemma 84).
In this thesis we will be concerned with finite groups with BN-pairs of subgroups
satisfying the axioms of [Tit1969]. Then is a normal subgroup of
and the Weyl group has a presentation with a set of distinguished involutionary
generators and defining relations
where is the order of in
and denotes a product of alternating
and with factors. The pair
is called a Coxeter system. The group is said to be of type
If we denote by
the least length of all expressions
(2.1.3) is called a reduced expression for if
There is a bijection between the double cosets and the elements
resulting in the Bruhat decomposition
The structure of the Hecke algebra of a
finite group with a BN-pair with respect to a Borel subgroup was shown in [Iwa1964] and [Mat1969-2] to be as follows.
Theorem (2.1.4)
has
where
with the identity element. Multiplication is determined by the formula
where the are the index parameters
For any reduced expression for
in
Thus is generated by
and has defining relations
where is as in (2.1.2).
Let be a Coxeter system and let
be indeterminates over chosen such that
if and only if and are conjugate in Let be
the polynomial ring
Then there exists an associative with identity, free basis
over and
multiplication determined by the formulas
(see [Bou1968], p. 55). The is called the generic ring corresponding
to the Coxeter system Analogous to Theorem (2.1.4) the generic ring
has a presentation with generators and relations
with is as in (2.1.2).
The Hecke algebra can be compared with the
group algebra as follows. Let be any field of characteristic zero and
a homomorphism. Consider as a
by setting
Then the specialized algebra
is an algebra over with basis
generators
and defining relations obtained from (2.1.8) by applying Thus if
is defined by
the index parameters (2.1.5), then
while if is defined by
for all then
We say the Coxeter system is of classical type if is of
type
or
In this chapter we will determine the irreducible representations of the generic
ring corresponding to a Coxeter system of classical type and by means of the appropriate specialized algebras the irreducible representations of the Hecke algebras
of groups with BN-pair of classical type.
The Representations of
If a Coxeter system is of type
is
isomorphic to the hyperoctahedral group, the group of signed permutations on letters (see 2). Thus
has a presentation with generators
where
and
the first sign change and relations
(see [Car1972]). Furthermore the set of generators is partitioned into 2 sets under conjugation; namely,
is conjugate to for while the negative
one-cycle is not conjugate to any
For the Coxeter system
taken as above, we take the generic ring to be defined
over the polynomial ring
indeterminates over It has a presentation with
generators
and relations
(B1)
(B2)
(B3)
(B4)
We depart from the notations of 1 strictly for notational convenience, i.e., we switch from
to
to avoid carrying around subscripts.
We now construct for each double partition
of a
of
The method involves the
construction of matrices over
for each of the generators of
in a manner analogous to the construction of the matrices of the transpositions
for the outer tensor product representation
of
For any integer let
Denote by the
matrix
Then
so the characteristic polynomial of gives
the identity matrix.
For let
Denote by
the matrix
As is obtained from
by setting
(2.2.1) shows
Denote by the
diagonal matrix
Then
In what follows, we employ the definitions and notations of (1) in regards to double partitions, standard tableaux, and axial distance.
Definition (2.2.6)
Let be a double
partition of and let
be the ordering of the standard tableaux of shape
according to the last letter sequence. Construct
matrices
over
as follows:
(1)
Construct by placing
(i)
in the entry if the letter
appears in of
(ii)
in the entry if the letter
appears in of
(iii)
zeros elsewhere.
(2)
Construct
by placing
(i)
in the entry if the letters
and appear in the same row of
or
(ii)
in the
entry if the letters and appear in the same column of
or of
(iii)
the matrix in the
and
entries corresponding to the tableaux
and where
(a)
and the letters and appear either both in
or of
(b)
is the axial distance from to in
(iv)
the matrix in the
and
entries corresponding to the tableaux
and where
(a)
and the letters and appear in different tableaux of
(b)
is the axial distance from to in
(v)
zeros elsewhere.
Let denote the free generated by
corresponding to the standard tableaux
of shape ordered according to the last letter sequence. For any field of
characteristic zero set
The corresponding basis elements of
will be denoted simply by Set
Define linear operators
on such that the matrix of with
respect to the basis
of is given by
Theorem (2.2.7)
Let and let
denote the generic ring of the
Coxeter system as
before. Let be a double partition of
Then the
map
defined by
is a representation of
Proof.
We need to show the relations (B1 - B5) are satisfied with in place of
We argue by induction on
For it is a case by case verification. The double partitions
and
are clearly seen to yield the well known one-dimensional representations of
([CIK1971], 10). For the double partition
there are two standard tableaux,
From (2.2.6)
and
Direct computation verifies the relation
Thus the relations (B1 - B3) are satisfied with and
in place of and
by the above computation, (2.2.2) and (2.2.5).
Now let
be a double partition of Deletion of the letter from a standard tableau automatically
yields a standard tableau involving letters. In fact deletion of from all standard tableaux having
at the end of the column will yield all standard tableaux of shape
Denoting this partition by and using the fact that all standard tableaux
with in the row precede all tableaux with in the
row for when ordered according to the last letter sequence, we have
and the corresponding matrix block form
as by (2.2.6), depends only on the letters
and It is understood that
is taken to equal zero if cannot appear in the row and the above summation, here and
elsewhere, will be taken over those which are non-zero. By the induction
hypothesis it therefore suffices to check the relations (B1 -B5) as they pertain to
The matrix from (2.2.6) is composed of the matrices
and
centered about the diagonal along with diagonal
entries and . Thus the relation
follows from (2.2.2), (2.2.4) and (2.2.5).
Let denote the subspace of
with basis
corresponding to the standard tableaux of shape with the letter appearing in the
row and appearing in the
row, the ordering of the basis taken according to the last letter sequence. Then
the summation taken over all allowable such that appears in row
and appears in row and this decomposition is consistent with the last
letter sequence arrangement of the basis of Thus, whenever
and are in distinct rows and columns, we have
as
for appearing in row appearing in row
where
Suppose first that and appear in distinct rows and columns, in the tableaux corresponding to
in row
in row Then
appears in row and appears in row in the tableaux
corresponding to and the map
gives an isomorphism
as
as the configuration of the first letters in the tableau corresponding to
is the same as the configuration of the first letters in the tableau corresponding to
In particular the matrix of
on is
where is the matrix of
on On the other hand, the matrix of
on
is, by (2.2.6), or
the axial distance from
to Thus the matrix of
on
is
where
or the
identity matrix. Then
for and (B5) holds. The only
other possibility is when and appear in the same row or column of the tableaux corresponding to
But in this case the matrix of
on is the scalar matrix
or by (2.2.6) and thus commutes with
on
This proves (B5) for all cases.
To check the relation (B4), we consider the restriction of
and to subspaces with basis
corresponding to all tableaux having a fixed arrangement of the first
letters and all possible rearrangements of the letters
and Let
denote the corresponding decomposition of the ordering of the basis of each
taken with respect to the last letter sequence. Then each is invariant
under and
and it suffices to check (B4) for the various possible arrangements of the last 3 letters in a case by case basis.
In what follows,
or will denote the matrix of
on
Case 1 — the letters and
in the same row or column. Then is one dimensional and
or by (2.2.6). Thus
and (B4) is satisfied.
Case 2 — the letters and
in two adjacent rows and two adjacent columns of the same diagram. Then is two-dimensional
with basis elements corresponding to tableaux where the configuration of the last 3 letters is
ordered according to last letter sequence. Then by (2.2.6),
and
in (a) and
and
in (b). Thus (B4) is satisfied in both cases by direct verification of the relation
Case 3 — the letters and
in two rows and three columns or three rows and two columns. Then is three dimensional with
basis elements corresponding to tableaux where the configuration of the last 3 letters is one of
ordered according to the last letter sequence. If we set
we have, by (2.2.6), in case (a),
and where
or , and
Here is the axial distance form
to in 2 and is the axial
distance from to in 1 so that
In case (b), and
with the same entries
in as in case (a). The analysis of (c) and (d) is similar except that now
in and Thus in all cases (B4) is satisfied by
Lemma (2.2.9)
Let be as above and let
Then for
(i)
and
(ii)
and
we have
Proof.
Observe that iff
(1)
(2)
(3)
(4)
(5)
For (1),
Now for
and for
Hence (2.2.10) equals zero for
and for
The relation (2) is entirely similar.
For (3) and (4) we have
and
But for and
For (5), first note the useful factorization
Now
But
for and for
So for both cases, (2.2.12) equals
using (2.2.11) with
Case 4 — the letters and
in three distinct rows and three distinct columns. Then is 6-dimensional with basis elements
corresponding to tableaux where the configuration of the last 3 letters is
ordered according to the last letter sequence. Let
Then if all rows are in the same diagram we have by (2.2.6),
and where
and
Here is the axial distance from to
in 1, is the axial distance from to
in 3 and is the axial distance from to
in 5 so that and all If two rows are in one diagram and the third in
the second diagram, we assume, without loss of generality, the lowest box to be in the second diagram. Superimposing the second diagram upon the first again does not
alter the relation except that now
only In this case
and
where now
and
Thus for both cases (B4) is satisfied by
Lemma (2.2.13)
Let and be as above and let
with Then for
(i)
(ii)
(iii)
we have
Proof.
First, either (i) or (ii) clearly imply (iii). Now
iff
(1)
(2)
(3)
(4)
(5)
(6)
and as these relations are symmetric in the and
it suffices to prove the lemma for (i).
For (1), (2), and (3), we observe that
Also set
We then obtain
For (4) we have
from (2.2.11), setting
The relations (5) and (6) are handled in an entirely similar manner.
This completes the proof of the lemma and the proof of the theorem.
Theorem (2.2.14)
Let be as before. The representations of
are irreducible,
pairwise inequivalent and are, up to isomorphism, a complete set of irreducible, inequivalent representations of
In particular is a splitting field for
Proof.
By induction in For the representations of
it is a matter of direct
computation to check irreducibility and inequivalence. Consideration of degrees shows a complete set of inequivalent representations is obtained. For
we employ the decomposition
(2.2.8) afforded by the last letter sequence and the position of the letter in a standard tableau. Let
be a double partition of The
is either irreducible or (2.2.8) is the decomposition of
into irreducible inequivalent
components, inequivalent because each of the double partitions of
is distinct. But for each pair
there exists a tableau with in row
in row and
a tableau with in row and in row
Thus the action of
does not decompose with respect to the
Hence is irreducible. Furthermore the double partition
is completely determined by the set of double partitions
of
Thus by the induction hypothesis
as
implies
From
([You1929]),
a double partition of Thus
is semisimple and as the
are defined over is
a splitting field for
This completes the proof.
It is clear that the above representations of the generic ring yield representations of a wide variety of specialized algebras of
Specifically, set
Corollary (2.2.15)
Let be any field of characteristic zero, a
homomorphism such that
Let be a double partition of and let
denote the linear operator on
obtained by the substitution
in the entry of
Then
is well defined and the
map
defined by
is a representation of
The representations
are a complete set of irreducible inequivalent representations of
Proof.
If
(2.2.1) and (2.2.3) show the matrices and
are well defined under the substitution
for
It is clear from the definition that axial distance in a Young diagram corresponding to a double partition of cannot exceed
in absolute value. Thus by (2.2.6), is well defined for
all As
has a presentation with generators and relations
obtained from (B1 - B5) by applying the proofs of Theorem (2.2.7) and (2.2.14) carry over to this case.
Let be a separable algebra over a field and let
be an algebraic closure of Define the numerical invariants of to be the set of integers
such that
is isomorphic to a direct sum of total matrix algebras
Thus for defined as in Corollary (2.2.15) the algebras
and have the same numerical invariants. In particular Corollary
(2.2.15) gives the well known result (see [BCu1972]) that for a finite group with BN-pair with Coxeter system
of type
Indeed in ([BCu1972]) this is shown to be the case for all Coxeter system with the possible exception of of type
Finally we remark that, for the specialization
defined by
the representations
are the irreducible representations of given by Theorem (1.2.3).
The Representations of and
We now obtain the representations of the generic ring of a Coxeter system of type and
If is a Coxeter system of type
is isomorphic to the symmetric
group and we take the set to be
where
We take the generic ring
to be defined over the
polynomial ring It has a presentation with
generators
and relations (B2, B4, B5).
Set The representations of
are readily obtained from the results of the previous section. As the matrices
are defined in (2.2.6) shows the matrices
are defined in
and
can be regarded as a linear operator on Thus
Theorem (2.3.1)
Let be a partition of and
The
map
defined by
is a representation of
The representations are a complete set of
irreducible, inequivalent representations of
Proof.
Theorem (2.2.7) shows the are representations of
Irreducibility and inequivalence follows from Theorem (2.2.14) as the matrix of
on is the scalar matrix
As (see [You1929])
the are a complete set of inequivalent representations and are absolutely irreducible.
The representations of the specialized algebra are handled entirely analogous to Corollary (2.2.15). Set
Then from the above and Corollary (2.2.15) we have
Corollary (2.3.2)
Let be any field of characteristic zero,
a homomorphism such that
Then for a partition of
the linear operators
are well defined and the
maps
defined by
is a representation of
The
are a complete set of irreducible, inequivalent representations of
Thus for as above the algebras
and
have the same numerical invariants. We remark that for the specialization the definitions of the matrices
shows the semi-normal matrix representation of
is obtained (see Theorem (1.2.1)).
If is a Coxeter system of type
can be regarded as a subgroup of index 2 in
acting on an orthonormal basis of
by means of permutations and even sign changes. A set of distinguished generators for
can be obtained from the set
of given in section (1) by setting
and taking the set to be
(see [Car1972]).
Let
be defined by Then the
specialized ring
has basis
with relations obtained by applying to (B1 - B5). In particular
Set
As is reduced in
we have
by (2.1.8). Applying to (B1 - B5) it is readily seen that
(B'1)
(B'2)
(B'3)
As any reduced expression of
in the generators
is a reduced expression for in the generators
of the relations (B'1 - B'3) show the subring of
generated by
has free basis
As all the generators
are conjugate in the subring generated by
is isomorphic to the generic ring of a Coxeter system of type
Denote this subring by
Thus the representations of
given by Corollary (2.2.15) provide us with representations of
Young ([You1929]) showed the restrictions of the representations of
to corresponding to a double partition
of remain irreducible if
and decomposes into two irreducible
components when
We show that this holds true in a generic sense.
Recall that a standard tableau for the double partition
of is an ordered pair
Then the tableau
is a standard tableau of shape called the
conjugate tableau of Moreover the map
is a bijection from the standard tableaux of shape to the standard tableaux of
shape Take
If
is the arrangement of the standard
tableaux of shape according to the last letter sequence, order the tableaux
of shape according to the scheme;
precedes if precedes
in the last letter sequence. Call this the conjugate ordering of the tableaux of shape
Let denote the matrix
Lemma (2.3.3)
Let
denote the matrix of
with respect to the basis of
ordered according to the last letter sequence,
Then
is the matrix of
with respect to the conjugate ordering of the basis of
Thus the restrictions
of the representations and
to
are equivalent.
Proof.
Let
be the arrangement of the standard tableaux of shape according to the last
letter sequence. For fixed
write as the direct sum
of invariant subspaces
where is taken to have basis
if and appear in the same row or column of and
has basis
where
in the last letter sequence. Let
denote the corresponding decomposition of
where has basis $
corresponding to
and where the ordering of
is taken with respect to the conjugate ordering. We need to show that if the matrix of
on
is the matrix of
on
is
This is a simple case by case verification.
1. If and are in the same row or column of they are
likewise in and the lemma is shown for this case.
2. If and are in distinct rows and columns of the same tableau
or of
set
and take Then
in the arrangement according to the last letter sequence while
in the conjugate ordering. The axial distance, from to
is the same both in and
Thus from (2.2.6) the matrix of
on
is
while the matrix of on
is
as is required.
3. If and are in distinct tableaux of
set
and take Then
in both the ordering according to the last letter sequence and the conjugate ordering. If is the axial distance from
to in
is the axial distance from to in
Let denote the
matrix obtained from under substitution
Then (2.2.6) shows the matrix of
on is
while the matrix of on
is
Direct computation verifies the relation
as is required.
It remains to show the lemma for
As acts on the basis
of
by scalar multiplication, the decomposition of into
invariant subspaces as
above is valid for
as well. It is furthermore clear from (2.2.6) that the action of
differs from that of
only on the spaces where the letters and
appear in distinct tableaux of
In this case the matrix of
on and on
is Using (2.3.4), a simple
matrix calculation completes the proof for this case. This completes the proof of the lemma.
We define the conjugate ordering of the standard tableaux
of shape as follows. Set
All standard tableaux belonging to precede those belonging to
in the arrangement according to the last letter sequence. Rearrange the last
tableaux in the last letter sequence, i.e. those in as follows; for
in
precedes if
precedes in the last letter sequence arrangement of the tableaux in
Lemma (2.3.5)
Let
denote the matrix of
on with respect to the conjugate ordering of the basis
of
Set
Then
Proof.
If
is a double partition of contained in
then so is
as in the proof of Theorem (2.2.7). Thus we have the decomposition,
in the conjugate ordering,
of as
generated by
Thus Lemma (2.3.3) shows that for
is of the form
which is easily seen to commute with
Hence we need to show (2.3.6) only for
If the letters and appear in the tableau
of
the proof of lemma (2.3.3) shows the matrix of
on the subspaces with corresponding basis
or
if
is of the from (2.3.8) and the above reasoning applies. If the letters
and appear in distinct tableaux of
with then
Thus and we can choose
such that in the last letter sequence arrangement of the
tableaux belonging to Then
is the arrangement of the tableaux according to the conjugate ordering. Taking the same ordering of the corresponding basis, the matrix of
on the subspace with
basis
is of the form
where
defined as in lemma (2.3.3) and the axial
distance from to in
A simple matrix calculation shows
This completes the proof.
Let
where for basis elements corresponding to tableaux
has basis
and
has basis
By Lemma (2.3.5) the maps
where the matrix of
with respect to the above basis is
are representations of
Theorem (2.3.9)
For double partitions
and of
the representations and
are a complete set of irreducible, inequivalent representations of
Proof.
By induction on For it is a matter of direct
verification. For the induction assumption and the proof of Lemma (2.3.5) shows
generated by and
Thus
and
are irreducible and inequivalent. The argument employed in Theorem (2.2.14) suffices for the irreducibility and inequivalence of the
By (2.3.7)
as
Thus none of the are equivalent to
Finally, consideration of degrees using the formula given in
Theorem (2.2.14) shows a complete set of inequivalent representations is obtained, and the representations are absolutely irreducible. Thus
is a splitting field for
This completes the proof.