Incidence Modules for Symplectic Spaces in Characteristic Two

0.28 MB·English

by  David B. Chandler

INCIDENCE MODULES FOR SYMPLECTIC SPACES IN
CHARACTERISTIC TWO
DAVID B. CHANDLER†, PETER SIN, QING XIANG‡
8
0
0 Abstract. We study the permutation action of a finite symplectic group of character-
2
istic2onthesetofsubspacesofitsstandardmodulewhichareeithertotallyisotropicor

n else complementary to totally isotropic subspaces with respect to the alternating form.
a
A generalformula is obtained for the 2-rank of the incidence matrix for the inclusion of
J
one-dimensional subspaces in the distinguished subspaces of a fixed dimension.
8
2

]
O 1. Introduction
C
Let V be a finite-dimensional vector space over F , q = 2t. We assume in addition that
. q
h
V has a nonsingular alternating form b( , ). Then the dimension of V must be an even
t
a · ·
number 2m. We fix a basis e ,e ,...,e ,f ,...,f of V, with corresponding coordinates
m 1 2 m m 1
x ,x ,...,x ,y ,...,y so that b(e ,f ) = δ , b(e ,e ) = 0, and b(f ,f ) = 0, for all i
[ 1 2 m m 1 i j ij i j i j
and j.

1
Forany subspace W ofV, let W = v V b(v,w) = 0, w W beitscomplement.
v ⊥
{ ∈ | ∀ ∈ }
2 A subspace W is called totally isotropic if b(x,x′) = 0 for any two vectors, x,x′ W. We
9 will be interested in totally isotropic subspaces of V; these necessarily have di∈mensions
3
r m. Also, we will consider the complements (with respect to the alternating form) of
4
≤
. totally isotropic subspaces, which of course will have dimensions r m.
1
≥
0 For 1 r 2m 1, let = (t) denote the set of totally isotropic subspaces, or
r r
8 ≤ ≤ − I I
complements of such, of dimension r. Let B = B (t) denote the (0,1)-incidence matrix
0 r,1 r,1
: of the natural inclusion relation between 1 and r. Its rows are indexed by r and its
v I I I
columns by , with the entry corresponding to Y and X equal to 1 if and
i 1 1 r
X I ∈ I ∈ I
only if Y is contained in X. We now state a theorem giving the 2-ranks of the matrices
r
a Br,1, that is, their ranks when the entries are considered as elements of F2. In order to
simplify this and several other statements we will employ the notational convention that
δ(P) = 1 if a statement P holds, and δ(P) = 0 otherwise.
Theorem 1.1. Let m 2 and 1 r 2m 1. Let A be the (2m r) (2m r)-matrix
≥ ≤ ≤ − − × −
whose (i,j)-entry is
2m 2m
a = .
i,j
2j i − 2j +i+2r 4m 2 2(m r)δ(r m)
(cid:18) − (cid:19) (cid:18) − − − − ≤ (cid:19)
Then
rank (B (t)) = 1+Trace(At).
2 r,1
Keywordsandphrases. Generalizedquadrangle,generallineargroup,2-rank,partialorder,symplectic
group, symplectic polar space.
†This author worked at Institute of Mathematics, Academia Sinica, Taipei, Taiwan.
‡Research supported in part by NSF Grant DMS 0701049.
1
2 CHANDLER,SIN,ANDXIANG
The significance of the entries a is that they are the dimensions of certain repre-
i,j
sentations of the symplectic group Sp(V) which are restrictions of representations of the
algebraic group Sp(2m,F ), where F is an algebraic closure of F .
q q q
Example 1.2. When m = r = 2, the matrix A is
4 4
,
1 5
(cid:18) (cid:19)
whose eigenvalues are 9 √17 = (1 √17)2. Thus,
± ±
2 2
2t 2t
1+√17 1 √17
rank (B (t)) = 1+ + − .
2 2,1
2 2
! !
This formula was previously proved in [9] by using very detailed information about the
extensions of simple modules for Sp(4,q); such cohomological information is unavailable
when V has higher dimension. Moreover, in [9] the numbers 1 √17 arise in a rather
±
2
mysterious fashion from the combinatorics involved. Theorem 1.1, whose proof does not
depend on [9], tells us that squares of these numbers are the eigenvalues of a matrix whose
entries are the dimensions of modules for Sp(4,q). This explanation is more natural.
We denote by k[ ] the space of functions from = to k = F . The action of Sp(V)
1 q
P P I
on makes k[ ] into a permutation module. Since Sp(V) acts transitively on , the
r
P P I
2-rank of B is equal to the dimension of the kSp(V)-submodule of k[ ] generated by
r,1
P
the characteristic function of one element of . Let us denote this submodule by .
r r
I C
Since the sum of all the characteristic functions of totally isotropic r-subspaces contained
in a given totally isotropic (r +1)-subspace is equal to the characteristic function of the
latter, with a similar relation for complements of totally isotropic subspaces, we have
= k[ ]. (1.1)
2m 1 2 1
C − ⊂ ··· ⊂ C ⊂ C P
Functions in k[ ] can be written as polynomials in the coordinates. For example, the
P
characteristic function of the isotropic subspace W defined by x = 0, x = 0, ..., x = 0
0 1 2 m
is
m
χ = (1 xq−1). (1.2)
W0 − i
i=1
Y
Theorem 1.1 is proved by relating this polynomial description to the actions of the groups
GL(V) and Sp(V) on k[ ]. We deduce it as a numerical corollary of Theorem 5.2 which
P
gives a basis of of the right size.
r
C
When q = pt is odd, the p-rank of the corresponding incidence matrices has been
computed in all dimensions in [4]. It was noted in [4] that for m = 2 and r = 2, there is
a single formula in p and t which gives the rank for both even and odd values of q, even
though there is no uniform proof. This circumstance is somewhat coincidental, however,
since the examples in Section 7 show that for m = r = 3, the ranks are not given by
the same function of p and t for the even characteristic and odd characteristic cases.
The even-odd distinction reflects some fundamental differences in the kSp(V)-submodule
structures of k[ ] between the even and odd cases, which are apparent in the action
P
INCIDENCE MODULES FOR SYMPLECTIC SPACES IN CHARACTERISTIC TWO 3
of Sp(V) on the graded pieces of the truncated polynomial algebra. In a sense which
will become clear, these modules are the basic building blocks of k[ ]. When p is odd,
P
the action of Sp(V) on each homogeneous piece of the truncated polynomial algebra is
semisimple, but for p = 2, the submodule structure is much richer. In characteristic 2 the
truncated polynomial algebra is an exterior algebra. The scalar extensions of the exterior
powers to the algebraic closure k are examples of tilting modules for the algebraic group
Sp(V k), as described in [6]; these have filtrations by Weyl modules and their duals. (See
⊗
[8] for definitions.) These filtrations are very important in our work; indeed the entries of
the matrix A in Theorem 1.1 are the dimensions of submodules which appear as terms in
them. In order to keep prerequisites to a minimum, our treatment of filtrations of exterior
powers in Section 4 is self-contained, requiring no general facts about Weyl modules, but
sufficient to define the filtrations and compute the dimensions of the subquotients.
The exterior powers are related to k[ ] through its kGL(V)-module structure. The
P
kGL(V)-submodule lattice of k[ ] has a simple description [1] in terms of a partially
P
ordered set of certain t-tuples of natural numbers. Each t-tuple in (0,0,...,0)
H H∪{ }
corresponds to a GL(V) composition factor of k[ ] and each composition factor is isomor-
P
phic to the t-fold twisted tensor product of exterior powers. These items are discussed in
detail in Section 2.
In Section 3 we shall define a set of polynomials which maps bijectively to a basis of
k[ ]. It is from this special basis, whose elements will be called symplectic basis functions
P
(SBFs), that we will find subsets which give bases of the incidence submodules for all
r
C
r. Each SBF is a t-fold product
f = f f2 f2t−1, (1.3)
0 1 ··· t 1
−
where each “digit” f is the function from V to k induced by a homogeneous square-free
j
polynomial. Homogeneous square-free polynomials correspond bijectively to elements of
the exterior power of the same degree, and the factorization (1.3) is compatible with the
twisted tensor product factorization mentioned above. Thus each of these functions has a
well-defined -type. We also introduce the notions of the class and level of each “digit”
H
to further subdivide our set of special functions. Since the class and level are defined for
each digit they apply also to elements of exterior powers. In Section 4, we show that the
filtration of the exterior powers by levels consists of kSp(V)-submodules, and compute
the dimensions of the subquotients.
The proof of Theorem 5.2 is then given in a series of lemmas in Section 5. A modified
version of the proof is given in Section 6 to handle the q = 2 case. Theorem 5.2 tells us
that a basis of consists of all those SBFs of certain -types and whose digits satisfy
r
C H
certain conditions on their levels. Thus, using the dimension computations of Section 4,
we obtain Theorem 1.1 as a corollary of Theorem 5.2.
2. k[ ] as a permutation module for GL(V)
P
In this section, V is a 2m-dimensional vector space over k = F , q = 2t. For the time
q
being, we do not equip V with an alternating form. So we simply consider k[ ] as a
P
kGL(V)-module. Let k[X ,X ,...,X ] denote the polynomial ring, in 2m indetermi-
1 2 2m
nates. Since every function on V is given by a polynomial in the 2m coordinates x , the
i
map X x defines a surjective k-algebra homomorphism k[X ,X ,...,X ] k[V],
i i 1 2 2m
7→ →
4 CHANDLER,SIN,ANDXIANG
with kernel generated by the elements Xq X . Furthermore, this map is simply the
i − i
coordinate description of the following canonical map. The polynomial ring is isomorphic
to the symmetric algebra S(V ) of the dual space of V; so we have a natural evaluation
∗
map S(V ) k[V]. This canonical description makes it clear that the map is equivariant
∗
→
with respect to the natural actions of GL(V) on these spaces. A basis for k[V] is obtained
by taking monomials in 2m coordinates x such that the degree in each variable is at most
i
q 1. We will call these the basis monomialsof k[V]. Noting that the functions on V 0
− \{ }
which descend to are precisely those which are invariant under scalar multiplication by
P
k , we obtain from the monomial basis of k[V] a basis of k[ ],
∗
P
2m
xbi 0 b q 1, b 0 (mod q 1), (b ,...,b ) = (q 1,...,q 1) .
{ i | ≤ i ≤ − i ≡ − 1 2m 6 − − }
i=1 i
Y X
We refer the elements of the above set as the basis monomials of k[ ].
P
2.1. Types and -types. We recall the definitions of two t-tuples associated with each
H
basis monomial. Let
2m t 1 2m
f = xbi = − (xaij)2j, (2.1)
i i
i=1 j=0i=1
Y YY
be a basis monomial of k[P], where bi = tj−=10aij2j and 0 ≤ aij ≤ 1. Let λj = 2i=m1aij.
The t-tuple λλλ = (λ ,...,λ ) is called the type of f. The set of all types of monomials is
0 t 1
denoted by Λ. − P P
In [1], there is another t-tuple associated with each basis monomial of k[ ], which we
P
will call its -type. This tuple will lie in the set (0,0,...0) , where
H H∪{ }
= s = (s ,s ,...,s ) j,1 s 2m 1, 0 2s s 2m .
0 1 t 1 j j+1 j
H { − | ∀ ≤ ≤ − ≤ − ≤ }
The -type s of f is uniquely determined by the type via the equations
H
λ = 2s s , 0 j t 1,
j j+1 j
− ≤ ≤ −
where the subscripts are taken modulo t. Moreover, these equations determine a bijection
between the set Λ of types of basis monomials of k[ ] and the set (0,0,...0) . We
P H∪{ }
will consider as a partially ordered set under the natural order induced by the product
H
order on t-tuples of natural numbers.
2.2. Composition factors. Thetypes, orequivalentlythe -typesparametrizethecom-
H
position factors of k[ ] in the following sense. The kGL(V)-module k[ ] is multiplicity-
P P
free. We can associate to each -type s (0,0,...0) a composition factor, which
H ∈ H∪{ }
we shall denote by L(s), such that these simple modules are all nonisomorphic, with
L((0,0,...,0)) = k. The simple modules L(s), s , occur as subquotients of k[ ] in
∼
∈ H P
the following way. For each s , we let Y(s) be the subspace spanned by monomials
∈ H
of -types in s = s′ s′ s , and Y(s) has a unique simple quotient, isomorphic
H H { ∈ H | ≤ }
to L(s).
The isomorphism type of the simple module L(s) is most easily described in terms of
the corresponding type (λ ,...,λ ) Λ. Let Sλ be the degree λ component in the
0 t 1
truncated polynomial ring k[X ,X−,..∈.,X ]/(X2;1 i 2m). Here λ ranges from 0
1 2 2m i ≤ ≤
to 2m. The truncated polynomial ring is isomorphic to the exterior algebra (V ); so
∗
∧
INCIDENCE MODULES FOR SYMPLECTIC SPACES IN CHARACTERISTIC TWO 5
the dimension of Sλ is 2m . The simple module L(s) is isomorphic to the twisted tensor
λ
product
(cid:0) (cid:1)
Sλ0 (Sλ1)(2) (Sλt−1)(2t−1). (2.2)
⊗ ⊗···⊗
2.3. Submodule structure. The reason for considering -types is that they allow a
H
simple description of the submodule structure of the kGL(V)-module k[ ]. The space
P
k[ ] has a kGL(V)-decomposition
P
k[ ] = k Y , (2.3)
P ⊕ P
where Y isthekernel ofthemapk[ ] k, f 1 f(Q). ThekGL(V)-module
Y is anPindecomposable module whPose→compo7→siti|oPn|−factoQrs∈Pare parametrized by . Then
P P H
[1, Theorem A] states that given any kGL(V)-submodule of Y , the set of -types of its
P H
composition factors is an ideal in the partially ordered set and that this correspondence
H
is an order isomorphism from the submodule lattice of Y to the lattice of ideals in .
P H
The submodules of Y can also be described in terms of basis monomials [1, Theorem
P
B]. Any submodule of Y has a basis consisting of the basis monomials which it contains.
P
Moreover, the -types of these basis monomials are precisely the -types of the compo-
H H
sition factors of the submodule. Furthermore, in any composition series, the images of
the monomials of a fixed -type form a basis of the composition factor of that -type.
H H
3. symplectic basis functions
We now define a set of square-free homogeneous polynomials of degree λ which we
will use to construct symplectic basis functions (SBFs). These square-free homogeneous
polynomials are in the indeterminates X ,X ,...,X ,Y ,...,Y , and will have the form
1 2 m m 1
given in the following definition.
Definition 3.1. Suppose the set of integers 1,...,m is partitioned into five disjoint
{ }
sets, R, R, S, T, and U. We require R = R = ρ, and take the cardinalities of
′ ′
| | | |
the other sets to be σ, τ, and υ, respectively, such that 2ρ + 2σ + τ = λ. We further
impose a pairing between R = r ,...,r and R = r ,...,r , so that r is paired with
{ 1 ρ} ′ { 1′ ρ′} i
r , 1 i ρ. We write W = X Y , 1 i m, and write Z to denote either X or else
i′ ≤ ≤ i i i ≤ ≤ i i
Y . Then we shall say that a square-free homogeneous polynomial F belongs to the class
i
(ρ,σ,τ,υ) if it can be written as
ρ
F = (Wri +Wri′) Wi Zi.
i=1 i S i T
Y Y∈ Y∈
We denote by Pλ the set of polynomials of those classes (ρ,σ,τ,υ) with 2ρ+2σ+τ = λ
ℓ
and σ ℓ. Then σ will be called the level of F, and ℓ will be called the level of Pλ.
≤ ℓ
We now construct a basis for k[ ]. We begin with a basis for the space of square-free
P
homogeneous polynomials of degree λ in X ,Y ,...,X ,Y . Note that the linear span of
1 1 m m
Pλ is a subspace of the span of Pλ.
ℓ 1 ℓ
−
Definition 3.2. Let λ be a fixed integer, 0 λ 2m. Choose Bλ to be a linearly
≤ ≤ 0
independent subset of Pλ of maximum size. For 1 ℓ λ/2, suppose that Bλ has been
0 ≤ ≤ ℓ 1
−
6 CHANDLER,SIN,ANDXIANG
chosen. Then we choose P λ Pλ such that Bλ = Bλ P λ forms a basis for the span
′ℓ ⊂ ℓ ℓ ℓ 1 ∪ ′ℓ
of Pλ. Finally, let −
ℓ
= Bλ .
B ⌊λ/2⌋
0 λ 2m
≤[≤
Note that Bλ is a basis for the space of all square-free homogeneous polynomials of
λ/2
degreeλ. Thus⌊ f⌋ormsabasisofthespaceofsquare-freepolynomialsinX ,Y ,...,X ,Y .
1 1 m m
Polynomials of Bthe form = F F 2 F 2t−1, where F , 0 j t 1 form a basis
0 1 t 1 j
F ··· − ∈ B ≤ ≤ −
for polynomials in these indeterminates, where the exponent of each indeterminate is at
most q 1.
−
The evaluation map φ : k[X ,...,X ,Y ,...,Y ] k[V], sending X to x and Y
1 m m 1 i i i
→
to y , 1 i m, restricts to a bijection from the space of square-free homogeneous
i
≤ ≤
polynomials in X ,...,X ,Y ,...,Y to those in x ,...,x ,y ,...,y . If F is a square-
1 m m 1 1 m m 1
free polynomial belonging to class (ρ,σ,τ,υ), then we shall refer to the function φ(F) as
belonging to that class also.
Definition 3.3. Let (s ,...,s ) and let f = f f 2 f 2t−1 k[ ] such that
0 t 1 0 1 t 1
the degree of f is λ = 2s − s ∈(sHubscripts modulo t),·a·n·d −f φ∈(BλjP ), for each
j j j+1 − j j ∈ λj
⌊ 2 ⌋
j, 0 j t 1. Then we shall say that f is a symplectic basis function (SBF), of the
≤ ≤ −
specified -type.
H
3.1. Linear substitutions in factorizable functions. We will frequently be dealing
with functions, such as basis monomials or SBFs, which can be factorized as a product
f = f f 2 f 2t−1 k[ ] (3.1)
0 1 t 1
··· − ∈ P
of functions f k[V] which are images under φ of homogeneous polynomials. We wish
j
∈
to think of f as the jth digit of f. However the representation (3.1) is not unique in
j
general; f = x2 can be factorized by taking f = x2 or by taking f = x . Thus when
1 0 1 1 1
we refer to the digit of such a function, it must be with a particular factorization in
mind. If we impose the additional requirement that the f in (3.1) be square-free, then
j
the digits are determined up to scalars and such a function f has a well-defined -type,
H
as in the case of basis monomials and SBFs. The more general notion of digit is needed
to discuss transformations of functions. Let b = b b 2 b 2t−1 be a function which has
0 1 t 1
··· −
a well-defined -type. An element g GL(V) acts on k[V] as a linear substitution, and
H ∈
we have
gb = (gb )(gb )2 (gb )2t−1. (3.2)
0 1 t 1
··· −
The digits gb in (3.2) are the homogeneous polynomials obtained from the b through
j j
linear substitution, andmay contain monomials with square factors. Like any function, gb
can be rewritten as a linear combination of factorizable functions with square-free digits,
such as basis monomials. We want to take a closer look at this rewriting process, keeping
track of the relationship between the -type of b and the -types of the terms in the
H H
possible rewritten forms of gb. The rewriting as a combination of basis monomials is done
as follows. The product (3.2) is first distributed into a sum of products of monomials,
each monomial having the factorized form (3.1), possibly with square factors in the digits.
Then if the first digit has a factor x2, the factor x2 is replaced by 1 and the second digit
i i
INCIDENCE MODULES FOR SYMPLECTIC SPACES IN CHARACTERISTIC TWO 7
is multiplied by x . We will call this a carry from the first digit to the second. Note
i
that this process causes the degree of the first digit to decrease by 2 while that of the
second increases by one. We perform all possible carries from the first digit to the second,
leaving the first digit square-free. We then repeat the process, performing carries from
the second digit to the third, and so on. Carries from the last digit go to the first digit,
and after that a second round of carries from the first to second digits, second to third,
etc. may occur. This process terminates because each time we have a carry from the last
digit to the first, the total degree decreases by 2t 1. To refine this idea we define, for
−
0 e t 1, the eth twisted degree of (3.1) by
≤ ≤ −
t 1
−
deg (f) = 2[j e]deg(f ),
e − j
j=0
X
where [j e] means the remainder modulo t. The effect of a carry from the (j 1)th
− −
digit to the jth digit is to lower the jth twisted degree by one, leaving other twisted
degrees unchanged. For those functions f which have a well-defined -type the relation
H
between the -type (s ,...,s ) and the twisted degrees is simple: (2t 1)s = deg (f).
H 0 t−1 − e e
We conclude that in the process of rewriting gb, whenever a carry is performed on a
monomial the basis monomial which results will have a strictly lower -type than our
H
original function b.
4. Submodules of the exterior powers
The ring k[X ,...,X ,Y ,...,Y ]/(X2,Y2)m can be viewed as the exterior algebra
1 m 1 m i i i=1
(V ), with X and Y corresponding to the basis elements x and y of V respectively.
∗ i i i i ∗
∧
Therefore, we can think of elements of (V ) as square-free polynomials in the X and Y
∗ i i
∧
and identify λ(V ) with Sλ, for 0 λ 2m.
∗
∧ ≤ ≤
For each λ, we define Sλ to be the k-span of Pλ, which is also the k-span of Bλ. Then
ℓ ℓ ℓ
the levels of functions define a filtration of subspaces
0 Sλ Sλ = Sλ.
⊂ 0 ⊆ ··· ⊆ ⌊λ2⌋
Note that if λ m then Sλ = 0 for ℓ < λ m.
≥ ℓ −
Lemma 4.1. Each Sλ is a kSp(V)-submodule of Sλ.
ℓ
We delay the proof of Lemma 4.1 until after the similar proof of Lemma 5.3.
Lemma 4.2. Assume λ m. Then Sλ has dimension 2m 2m . A basis of Sλ consists
≤ 0 λ − λ 2 0
of all elements of P0λ of the form ρi=1(Wri + Wri′) (cid:0)t T(cid:1)Zt,(cid:0)w−he(cid:1)re R = {r1,r2,...,rρ},
R = r ,r ,...,r and T are disjoint subsets of 1,...∈,m such that 2 R + T = λ and
′ { 1′ 2′ ρ′} Q { Q } | | | |
the following conditions hold.
(1) r < r < < r and r < r < < r .
1 2 ··· ρ 1′ 2′ ··· ρ′
(2) r is the smallest element of 1,...,m T which is not in the set r ,r j < i .
i { }\ { j j′ | }
Proof. A proof can be found in [2], Theorem 1.1; an earlier proof appears in the examples
of [7] p. 39. (We only need the statements in the characteristic 2 case, but these references
(cid:3)
treat all characteristics.)
8 CHANDLER,SIN,ANDXIANG
Remark 4.3. The module Sλ or, more precisely, their scalar extensions to an algebraic
0
closure k are examples of Weyl modules for the algebraic group Sp(k V) and it is in
k
⊗
this context that they were studied in [7] and [2]. We refer the interested reader to [8] for
the definitions and properties of this important class of modules.
Lemma 4.4. Assume λ m.
≤
1. We have an isomorphism
Sλ/Sλ = Sλ 2ℓ. (4.1)
ℓ ℓ 1 ∼ 0−
−
2. The dimension of Sλ is 2m 2m .
ℓ λ − λ 2ℓ 2
− −
Proof. We shall define a lin(cid:0)ear(cid:1)map(cid:0) (cid:1)
α : Sλ 2ℓ Sλ.
0− → ℓ
Let f be a basis element of Sλ 2ℓ given by Lemma 4.2, with index sets (R,R,T). Set
0− ′
M = 1,...,m . Then the number of indices involved in f is 2ρ+τ = λ 2ℓ, and the
{ } −
number of unused indices is (m λ) + 2ℓ 2ℓ. We choose an arbitrary ℓ-subset S of
− ≥
these unused indices and define
α(f) = f W .
s
s S
Y∈
Composing with the natural projection gives a map
α : Sλ 2ℓ Sλ/Sλ .
0− → ℓ ℓ 1
−
We claim that the map α is surjective.
Now Sλ is spanned by polynomials of the form
ℓ
(Wri +Wri′) WSi Zti, (4.2)
ri∈RY,ri′∈R′ sYi∈S tYi∈T
where R, R, S and T are disjoint subsets of M = 1,...,m , with R = R , and
′ ′
{ } | | | |
2 R + 2 S + T = λ, and S ℓ. The classes of these elements such that σ = ℓ span
| | | | | | | | ≤
the quotient space Sλ/Sλ . We consider what restrictions may be placed on the form of
ℓ ℓ 1
such a polynomial and st−ill have a spanning set. Of course we may assume that the r
i
form an increasing sequence and that r < r for all i. Furthermore, the equation
i i′
(W +W )(W +W ) = (W +W )(W +W )+(W +W )(W +W ) (4.3)
1 4 2 3 1 3 2 4 1 2 3 4
implies that we can assume that the r also form an increasing sequence. Next observe
i′
that
(W +W ) = (W +W )+(W +W ) (4.4)
2 3 1 2 1 3
and
(W +W )W = (W +W )W +(W +W )W . (4.5)
2 3 1 1 3 2 1 2 3
Equations (4.4) and (4.5) respectively allow us to assume in addition that no element
of U or S precedes any element of R, or in other words that for each i, r is the smallest
i
index in ( 1,...,m T) r ,r j < i . Thus the sets R and R can be assumed
{ } \ \ { j j′ | } ′
to satisfy the same conditions as the corresponding sets for the basis elements of Sλ 2ℓ
0−
stated in Lemma 4.2. We have proved that there is a spanning set of Sλ such that for
ℓ
each element for which σ = ℓ, the subsets R and R and T are exactly the same as those
′
INCIDENCE MODULES FOR SYMPLECTIC SPACES IN CHARACTERISTIC TWO 9
of some α(f), where f is a basis element of Sλ 2ℓ. Finally, we note that two elements
0−
of Pλ with the same sets R, R and T are equal modulo Sλ . To see this, suppose the
ℓ ′ ℓ 1
index sets of the two elements are (R,R,S,T,U) and (R,R−,S ,T,U ) and assume that
′ ′ ∗ ∗
the symmetric difference of S and S is s,s . Then the sum of the two polynomials has
∗ ∗
{ }
index sets (R s ,R s ,S S ,T,(U U ) s,s ); so it belongs to Sλ . Therefore
∪{ } ′∪{ ∗} ∩ ∗ ∪ ∗ \{ ∗} ℓ 1
the map α is surjective (and, incidentally, independent of the choice of ℓ-su−bset S in the
definition of α).
Thus, since Lemma 4.2 gives the dimension of Sλ 2ℓ, we have
0−
λ
2m ⌊2⌋
= dimSλ = dim(Sλ/Sλ )
λ ℓ ℓ 1
−
(cid:18) (cid:19) ℓ=0
X
λ
⌊2⌋
dimSλ 2ℓ
≤ 0−
ℓ=0 (4.6)
X
λ
⌊2⌋ 2m 2m
=
λ 2ℓ − λ 2ℓ 2
ℓ=0 (cid:18) − (cid:19) (cid:18) − − (cid:19)
X
2m
= ,
λ
(cid:18) (cid:19)
(cid:3)
so equality holds throughout. Both parts of the lemma now follow.
The case where λ m will be treated by using various dualities, which we now proceed
≥
to discuss. We do not assume λ m yet. We start with the kGL(V)-isomorphism
≥
λ(V ) = ( λ(V)) . Let λ = 2m λ. Then there is a natural pairing
∗ ∼ ∗ ∗
∧ ∧ −
λ(V) λ∗(V) 2m(V) = k
∧ ×∧ → ∧
givenbyexteriormultiplication, whichdefinesakGL(V)-isomorphism( λ(V)) = λ∗(V).
∗ ∼
∧ ∧
Finally, the kSp(V)-isomorphism V = V given by v b(v, ) induces a kSp(V)-
∼ ∗
isomorphism λ∗(V) = λ∗(V ). Combining these isomorph7→isms yie·lds a kSp(V)-module
∼ ∗
∧ ∧
isomorphism
Sλ Sλ∗, X Y X Y ,
I J M J M I
→ 7→ \ \
for each λ. Putting these isomorphisms together for all λ yields a kSp(V)-automorphism
δ of (V ) of order 2. A straightforward calculation shows that a basis polynomial of
∗
∧
degree λ and class (ρ,σ,τ,υ) is mapped under δ to a basis polynomial of degree λ and
∗
class (ρ,υ,τ,σ). (Actually, the map δ is very simple; a basis polynomial of Sλ∗ with index
sets (R,R,S,T,U) is mapped to basis polynomial of Sλ with index sets (R,R,U,T,S); δ
′ ′
just swaps U and S.) Now assume λ m. Since m = 2ρ+σ+τ+υ and λ = 2ρ+2σ+τ,
∗
≥
we have
υ = m λ +σ = (λ m)+σ,
∗
− −
which shows that
δ(Sλ∗) = Sλ ,
ℓ∗ (λ m)+ℓ∗
−
for 0 ℓ λ∗ .
≤ ∗ ≤ ⌊ 2 ⌋
10 CHANDLER,SIN,ANDXIANG
We have already computed the left hand side, so if we set ℓ = (λ m)+ℓ then
∗
−
2m 2m 2m 2m
dimSλ = = .
ℓ λ − λ 2ℓ 2 λ − λ 2ℓ 2
(cid:18) ∗(cid:19) (cid:18) ∗ − ∗ − (cid:19) (cid:18) (cid:19) (cid:18) − − (cid:19)
We have proved:
Theorem 4.5. Let 0 λ 2m and max 0,λ m ℓ λ . Then Sλ has dimension
≤ ≤ { − } ≤ ≤ ⌊2⌋ ℓ
2m 2m
.
λ − λ 2ℓ 2
(cid:18) (cid:19) (cid:18) − − (cid:19)
Remark 4.6. The isomorphism in Lemma 4.4 is in fact one of kSp(V)-modules, and
after extending the field to k it is an isomorphism of rational Sp(V )-modules. In fact,
k
it can be shown that the filtration given by the submodules k Sλ is the filtration of
⊗k ℓ
λ(V ) by Weyl modules, dual to the good filtration described in [6], Appendix A, p.71
∗
∧ k
(for all characteristics).
5. Bases for incidence modules
In this section we assume that q > 2. In the next section we show that Theorem 5.2
below also holds for q = 2, with a slightly different proof.
Definition 5.1. Let 1 r 2m 1 and let f = f f 2 f 2t−1 k[ ] with f
0 1 t 1 j
≤ ≤ − ··· − ∈ P
homogeneous and square-free for all j. If the -type of f is (s ,s ,...,s ), let
0 1 t 1
H −
ℓ = (m r)δ(r m)+(2m r s ). (5.1)
j j
− ≤ − −
The function f is called r-admissible if f is a constant function, or if
(a) for each j, 0 j t 1, f φ(Pλj), and
≤ ≤ − j ∈ ℓj
(b) the -type of f is less than or equal to (2m r,2m r,...,2m r).
H − − −
In particular, if f is constant or if f φ(Bλj) for all j, 0 j t 1, we call f an
j ∈ ℓj ≤ ≤ −
r-admissible SBF.
Note that by construction, r-admissible functions are in the linear span of r-admissible
SBFs.
Theorem 5.2. Assume q > 2. The r-admissible symplectic basis functions form a basis
for the kSp(V)-submodule of k[ ].
r
C P
The rest of this section is devoted to proving this theorem.
Let denote the linear span of the r-admissible SBFs, for 1 r 2m 1.
r
M ≤ ≤ −
λ
Let P = φ(Pλ).
ℓ ℓ