Extensions of strict polynomial functors
Marcin Cha_lupnik *
Instytut Matematyki, University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland
e-mail: mchal@mimuw.edu.pl
Abstract
We compute Ext-groups between Frobenius twists of strict polyno-
mial functors. The main results are calculations of Ext*P(Dd(i), F (i))
and Ext*P(W~(i), Sd(i)~), where Dd is the divided power functor, W~
and S~ are respectively the Weyl and Schur functors associated to
diagrams ~, ~ of the same weight, and F is an arbitrary functor.
1 Introduction
Computing Ext-groups between GLn(k)-modules for a field k of positive
characteristic had been known to be a very difficult problem for a long time.
Only recently, introducing a suitable category of functors F in [HLS] has
changed the situation significantly. As it was demonstrated in [FLS], [FS],
[FFSS], it is possible to make effective computations of Ext-groups in the
category F (and its more sophisticated modification P ). In a meantime, in
a series of papers ([B2],[K1],[K2],[K3], [FS], [FFSS]) there was established
a close relation between the functor categories F, P and the category of
______________________________
*The author was partially supported by Bia_lynicki-Birula Subsydium of Found*
*ation of
Polish Science and the polish scientific grant (KBN) 2 P03A 002 18.
1
GLn(k)-modules. In particular, it was shown in [B2] and [FFSS] that there
is an isomorphism of Ext-groups
Ext*P(F (i), G(i)) ' Ext*GLn(k)-mod(F (kn), G(kn)),
for a finite field k and integers i, n large enough (F (i)means the i-th Frobe-
nius twist of a functor F (cf. [FS], sect. 1)). Thus in order to get calculatio*
*ns
for GLn-modules one should compute Ext-groups in the category P between
twists of functors. Such calculations were started in [FS] where (adapting
ideas of [FLS] to the context of the category P ) the groups Ext *P(I(i), I(i))
were computed. Already this result has a valuable application to the GLn(k)-
modules, for after some additional work ([FS], sect. 7) it leads to a computa-
tion of H*(GLn(Fp), Mn(Fp)) for large n (with action of GLn(Fp) on matri-
ces by conjugation), which is a difficult still unpublished result of Bökstedt
([Bö]). These computations were extended in ([Fr],[FFSS],[PS]).
The aim of the present paper is to generalize and systematize computa-
tions of Ext-groups in the functor category by using methods of representa-
tion theory. We obtain complete description of the Ext-groups for a large
class of functors strongly generalizing and putting into a uniform context
known computations.
A direct inspiration for this work was a computation of Ext*P(Dd(i), Sd(i))
obtained in [FFSS]. Since the tensor products of divided powers form a family
of projective generators of P and the products of symmetric powers _ of
injective ones, one can hope for computations of Ext-groups for the Frobenius
twists of functors of a more general form.
The main results are: the computation of Ext*(Dd(i), F (i)) for arbitrary F
(Th. 4.3) and the computation of Ext*(F (i), Gd(i)) for F, G satisfying certain
simple abstract condition (Th. 4.4). The most important instance of Th. 4.4
is that for functors F = W~ and G = S~ (resp. Weyl and Schur functors)
for diagrams ~, ~ of the same weight. The language in which results of
computations are given utilizes a concept of "symmetrization of a functor"
(see sect. 3). This notion exploits a strong interplay between representations
of the general linear group and the symmetric group coming from the action
of these groups on the tensor power of a space, and may be thought of as a
generalization of a classical notion of symmetrization of a representation.
This paper is a first part of my work on homological algebra in the cate-
gory of functors. In the next article ([C2]) I partially expand computations of
2
Ext-groups between twisted Weyl and Schur functors to the case of diagrams
of different weights. As it is not surprising for a reader of [FLS], [FS], [FFS*
*S],
the essential role in that work is played by the De-Rham complex. Its ap-
propriate generalization to the case of an arbitrary Young diagram turned
out to be an object complicated and interesting for its own. I investigate it
in detail in a separate article ([C1]).
2 Recollections of diagrams and functors
We start by collecting some basic facts concerning Young diagrams and func-
tors one can associate to them. A Young diagram ~ of weight d is justPa
weakly decreasing sequence of positive integers (~1, . .,.~l) with lj=1~j =
d =: |~|. We can associate to a Young diagram ~ the conjugate diagram e~
whose rows are columns of ~ (formally: e~k= #{j : ~j k}). We will con-
sider the partial order of dominance on the set ofPYoung diagrams.PWe say
that ~ dominates ~ (~ x ~) if for all j we have i j~i i j~i. This par-
tial order may be enriched to the total lexicographic order: ~ ~ if for the
least i such that ~i 6= ~i, we have ~i > ~i. The direction of dominance and
lexicographic relations looks strange, since the lesser diagram is the longer
rows it has. The reason is that the terminology in my two main references:
[ABW] and [CPS] is not consistent. I decided to follow the conventions of
[ABW] when dealing with Schur functors etc, but I follow [CPS] with respect
to the direction of orders.
Given two diagrams ~ ~ (ie. ~j ~j for all j), we may form a skew
diagram ~=~ which should be imagined as a diagram ~ with deleted boxes
belonging to ~. Here is a picture for (4, 3, 1)=(2, 1)
_____
________||||
_______||||
___|||
Throughout this paper k will be a field of positive characteristic p and P d
will denote the category of homogeneous strict polynomial functors of degree
d over k (see [FS], sect. 2). All Ext-groups will be computed in P d for
appropriate d. We now recall certain important objects in P d. The most
3
fundamental are: the d-th tensor power Id(V ) := V d, the d-th symmetric
power Sd(V ) := (V d) d, the d-th divided power Dd(V ) := (V d) d, (the
last two functors are not isomorphic for d p) and the d-th exterior power
d(V ) := (V d) d ' (V d) d for the alternating action of d on the tensor
power (this definition needs a modification for p = 2, I will discuss it in det*
*ail
in the next section). There are also well known transformations between
these functors eg. the inclusion cd : d -! Id and the epimorphism md :
Id -! Sd. Given a diagram ~ of weight d, we put ~ := ~1 . . .~land
c~ := c~1 . . .c~l : ~ - ! Id. In the same fashion we define Se~and
me~: Id - ! Se~, but one should remember that me~acts in a öc njugate
manner" ie. we gather the elements which have indices belonging to the
same column (see [ABW], sect. II.1). We are now in a position to introduce
a more complicated object. The Schur functor S~ is defined as the image of
the composition me~O c~. It comes with two structural transformations: the
epimorphism OE~ : ~ -! S~ and monomorphism _~ : S~ -! Se~, which in
extreme cases give isomorphisms S(d)' d, S(1d)' Sd.
There is a useful contravariant duality in the category P d called the Kuhn
duality: F #(V ) = (F (V *))* where V *means the k-dual space. It is easy to
check that (Dd)# ' Sd while d is selfdual. We will also consider the Kuhn
duals of Schur functors which are called Weyl functors and denoted by W~.
The independent definition of Weyl functor is, of course, as the image of the
composition De~-! Id -! ~.
All these constructions may also be applied to skew diagrams. Although
skew Schur and Weyl functors play less important role in the theory (in fact
the Littlewood-Richardson rule ([Bo]) says that any skew Schur functor has
a filtration with a graded object being a sum of Schur functors), they are
often useful in inductive arguments.
We need two important homological properties of Weyl and Schur func-
tors.
Fact 2.1 For any skew diagrams ~=~0, ~=~0, Ext n(W~=~0, S~=~0) = 0,
for n > 0.
Fact 2.2 If for some n > 0, Ext n(S~, S~) 6= 0, then ~ * ~.
Fact 2.1 for solid (ie. not skew) diagrams is mentioned in the proof of ([CPS],
Th. 3.11). The general case follows immediately from the Littlewood-Richardson
rule.
4
Fact 2.2 is a part of ([CPS], Lemma 3.2). Both facts are purely formal
consequences of the axioms for a "highest weight category" ([CPS], Def. 3.1).
||
I would like to finish this section by introducing the main technical tool, whi*
*ch
will be used repeatedly in the next sections. This tool is the Decomposition
Formula. Let Pn denote the category of strict polynomial functors in n
variables. The Decomposition Formula ([ABW], Th. II.4.11) provides an
extremely useful filtration of a functor in two variables S~=~(V W ).
Fact 2.3 (Decomposition Formula) The bifunctor S~=~(V W ) has a
filtration Mff(V, W ) (for ff satisfying ~ ff ~) and an order in the filtra*
*tion
comes from the lexicographic order among ff. Its associated graded object is
M
Sff=~(V ) S~=ff(W ).
~ ff ~
Iterating this procedure we get a filtration of the n-functor S~=~(V1 . . .Vn).
Corollary 2.4 The functor in n variables S~=~(V1 . . .Vn) has a filtration
M~ ff1 ... ffn-1,~with ordering coming from the n-fold lexicographic order
(ie. to compare sequences (ff1, . .,.ffn-1) i (ff01, . .,.ff0n-1) we pick t*
*he
smallest i such that ffi 6= ff0iand compare lexicographically ffi and ff0i). I*
*ts
graded object is
M
Sff1=~(V1) . . .Sffn-1=ffn-2(Vn-1) S~=ffn-1(Vn).
~ ff1 ... ffn-1 ~
Of course, we get an analogous decomposition for twisted Schur functors and
for Weyl functors. This filtration is a powerful tool in computations of Ext-
groups, since as it was observed in ([FFSS], pp. 671-672), the evident adjoint
functors between P and Pn yield an isomorphism
Ext*P(F1 . . .Fn, S(i)~=~) = Ext*Pn(F1(V1) . . .Fn(Vn), S(i)~=~(V1 . . .Vn)*
*),
I write down spaces V1, . .,.Vn in the right-hand side of the formula to empha-
size the dependence of a functor on all n variables. We recall from ([FFSS],
pp. 672), that the üK nneth formula" gives an isomorphism
Ext *Pn(F1(V1) . . .Fn(Vn), S(i)ff1=~(V1) . . .S(i)~=ffn-1(Vn)) =
5
= Ext*P(F1, S(i)ff1=~) . . .Ext*P(Fn, S(i)~=ffn-1).
Thus the Decomposition Formula leads to a spectral sequence, which we will
call the Decomposition Spectral Sequence.
Corollary 2.5 There exists a spectral sequence converging to Ext*P(F1 . . .
Fn, S(i)~=~), whose E1-term has the form
M i (i) i (i)
E1ij= ExtP1(F1, Sff1=~) . . .ExtPn(Fn, S~=ffn-1),
i1+...+in=i+j
where j stands for a place of (ff1, . .,.ffn-1) in the (n - 1)-fold lexicograph*
*ic
order.
Analogous sequences also exist for Ext*P(F1 . . .Fn, W~(i)=~), Ext*P(S(i)~=~*
*, F1
. . .Fn) etc.
In the present paper we will mainly deal with a very special case of the
Decomposition Formula (already considered in [FFSS]), namely the one for
the diagram (1d). The Decomposition Formula in this case splits and takes
the form of the known formula
M
Sd(i)(V W ) = Sj(i)(V ) Sk(i)(W ).
j+k=d
Hence the Decomposition Spectral Sequence also splits and gives the formula:
i(i) * |Fn|=pi(i)
Ext*P(F1 . . .Fn, Sd(i)) = Ext*P(F1, S|F1|=p ) . . .ExtP (Fn, S ),
for any homogeneous functors F1, . .,.Fn. We get analogous formulae for the
divided and exterior powers and for products of homogeneous functors on
the second variable instead of the first. The Schur functors for which the De-
composition Formula takes this simplest form where investigate in detail in
[FFSS] where they were called "exponential functors" (see [FFSS], p. 670).
These particular instances of the Decomposition Formula and Decomposi-
tion Spectral Sequence will be referred to as the Exponential Formula. Some
more advanced applications of the Decomposition Formula and Decomposi-
tion Spectral Sequence will appear in [C1] and [C2].
6
3 Symmetrization of a functor
Let F d denote the category of additive k-linear functors from the cat-
egory of graded finitely generated d-modules to the category of graded
finitely generated k-spaces (k-linearity of a functor means that the struc-
tural map Hom k[ d](M, N) -! Hom k(f(M), f(N)) is k-linear; all mor-
phisms, actions etc. are assumed to preserve grading). We will call the
objects of this category d-functors and the morphisms d-transformations.
Observe that for any d-functor f, an assignment V 7! f(V d) (we re-
gard V as concentrated in degree 0) defines a homogeneous strict poly-
nomial functor of degree d. If so happens, we say that a d-functor is a
symmetrization of the respective strict polynomial functor. In fact, we of-
ten define strict polynomial functors just giving their symmetrizations, eg.
Sd = f(V d), for the d-functor f(M) = (M) d. Usually (if it causes no
confusion) we will denote the symmetrization of a functor by the same let-
ter but small. For example it is clear what we mean by s~, d~, s~, w~, eg:
s~(M) := im ((Malt) ~ - ! M -! (M) e~) (for any d-module M, Malt
stands for M sgn). It is also self-evident how these functors behave with
respect to the grading: degree of a tensor product is just a sum of degrees of
factors. One should be more cautious in the case of the exterior power for
two reasons. The first is that the invariants and coinvariants of the alternat-
ing action are not isomorphic d-functors in general. So we should distinct
between ~~inv(M) = (Malt) ~, and ~~coinv(M) = (Malt) ~, (by the way, both
d-functors are symmetrizations of the strict polynomial functor d). The
second reason is a pathology which happens for p = 2, when we cannot define
the exterior power as the (co)invariants of the alternating action. We will
briefly discuss a modification which is needed in definition of ~dinv(the argu-
ment for ~dcoinvis similar). We start with d = 2. Then we may define ~2inv
as the kernelTof the 2-epimorphism id -! s2. For an arbitrary d we define
~dinvto be 2 dker(id - ! id 2). The definition meets our expectations
because d is generated by the set of transpositions. Its main advantage is
that it referres only to a given action of the symmetric group. Therefore from
now on we will need not to consider the case p = 2 separately.
It is worth mentioning that the idea of symmetrization is present in
many constructions in representation theory. For example, applying cer-
tain d-functors to the d-bimodule k[ d] we obtain some important d-
modules (eg. s~(k[ d]) is so-called Specht module Sp~). Finally, observe
7
that we still have the Kuhn duality. Namely, for a d-functor f, we put
f# (M) := (f(M*))*, where * at the right-hand side means the k-linear du-
ality. Now it is easy to check that s#~= w~ and in particular ~~#inv= ~~coinv.
Later on we will focus on symmetrizations satisfying some additional techni-
cal condition.
Definition 3.1 A d-functor fin is called an injective symmetrization of a
functor F 2 P d if fin(V d) = F (V ), and there exists a d-transformation
k d
_ : fin -! ks~ such that _(V ) is an inclusion.
Similarly, we say that a symmetrization fpr is a projective symmetrization if
k pr d
there exists a d-transformation OE : kd~ -! f whose evaluation on V
is onto.
The importance of this class of symmetrizations comes from the fact that the
family {S~} (resp. {D~}) forms a set of injective (resp. projective) generators
of P d ([FS], Th. 2.10). In order to express concisely another important
property of injective symmetrizations we need the following definition.
DefinitionL3.2 We say that a d-module M is a Y-permutative module if
M ' ni=1Mi, where Mi = k k[Hi]k[ d] for some Young subgroups Hi (cf.
[JK], sect. 1.3).
The most important example of a Y -permutative module is a d-module
V d for any space V . Now we can go back to symmetrizations.
Fact 3.3
1. Any strict polynomial functor has an injective and a projective sym-
metrization.
_0 ~0
2. Let fin -! s be an injective symmetrization of F (I adopt the con-
0 ~0k
vention: s~ := ks ). Then _0 may be extended to a sequence of
d-transformations
_0 ~0 _1 ~1 _2 _l ~l
fin -! s -! s -! . .-.! s ,
such that for any Y -permutative d-module M, the sequence
_0 ~0 _1 ~1 _2 _l ~l
0 -! fin(M) -! s (M) -! s (M) -! . .-.! s (M) -! 0
is exact.
An analogous fact holds for a projective symmetrization.
8
Proof: We start with comparing transformations and d-transformations in
a very special case.
Lemma 3.4 For any diagrams ~, ~0 of weight d
0 ~ ~0
Hom Pd(S~, S~ ) = Hom F d (s , s ).
Proof: Since Hom P(Sk, Sk) = k, then applying the Exponential Formula to
0
both variables we get a description of Hom (S~, S~ ). From a purely com-
binatorial point of view we may describe it as a space having basis labeled
by matrices consisting of positive integers satisfying the following conditions:
each row is weakly decreasing, the sum of numbers in the ith row equals
~i, the sum of numbers in the ith column equals ~0j(cf. [FFSS], Cor. 1.8).
Looking at the construction of the Decomposition Formula it is easy to find
the transformation corresponding to a given element of the basis. Namely,
to a matrix [aij] we associate a composition
O O 0
S~ -! Saij' Saji- ! S~ ,
ij ij
where the first and third arrows are respectively tensor products of iter-
ated comultiplication and multiplication in the symmetric power, while the
second arrow interchanges factors which on the left-hand side are ordered
with respect to rows and on the right-hand side with respect to columns (cf.
[FFSS], pp. 673-676). Thus we see that any transformation is a composition
of transformations of three simple types (possibly tensored with identities):
the multiplication Sa Sb -! Sa+b, the comultiplication Sa+b -! Sa Sb,
and the transposition Sa Sb -! Sb Sa. These transformations, of course,
come from d-transformations, respectively from the induction, the restric-
tion and the homomorphism of the groups. Thus we have shown that any
transformation comes from some d-transformation.
0
It remains to show that a nontrivial d-transformation _ : s~ -! s~ has
0
a nontrivial evaluation _(V d) : S~ - ! S~ . It will be more convenient to
0 ~
work with the Kuhn dual of _, which is a d-transformation _# : d~ - ! d .
Suppose that im(_# )(V d) = 0. Then im(_# ) is a left exact functor vanish-
ing on all d-modules V d. But if dim (V ) = d then V d contains k[ d] as
a direct summand. Thus im (_# )(k[ d]) = 0. But since any finitely gener-
ated d-module embeds into a free module and im(_# ) preserves monomor-
phisms, it must be the trivial functor. ||
9
In order to construct an injective symmetrization of a strict polynomial func-
tor F we consider the beginning of a finite injective resolution of F by the
sums of products of symmetric powers
_00 ~0 _01 ~1
0 -! F - ! S -! S -! . . ..
The existence of such a finite resolution follows easily from the axioms for
a highest weight category ([CPS], Def. 3.1) and the Littlewood-Richardson
rule. Of course F = ker(_01). Thanks to Lemma 3.4 we know that the trans-
0 ~1
formation _01comes from the d-transformation _1 : s~ -! s . Therefore
the d-functor fin := ker(_1) is an injective symmetrization of F . This
finishes the proof of the first part of Fact 3.3.
To obtain the second part we take the whole resolution
_00 ~0 _01 ~1 _02 _0l ~l
0 -! F - ! S -! S -! . .-.! S -! 0.
According to Lemma 3.4 it lifts to the sequence of d-transformations
_0 ~0 _1 ~1 _2 _l ~l
0 -! fin -! s -! s -! . .-.! s - ! 0,
whose evaluation on V d is exact. The exactness of evaluation on an arbi-
trary Y -permutative module follows from the fact, that any such a module
is a direct summand in a finite sum of V d for a space V of dimension d. ||
Of course, an injective symmetrization is not unique. The exterior power
provides the easiest example since both ~dinv-! id and ~dcoinv-! id are its
injective symmetrizations. The point is that however the arrow ~dcoinv-! id
(ä veraging to invariants") is not monomorphic but its evaluation on V d
is, which is sufficient. For a similar reason s~ is not only injective but also
projective symmetrization of S~ (an analogous fact holds for Weyl functors).
We finish this section with one more tricky example of an injective sym-
metrization. We shall find an injective symmetrization of Sd(1). To do this
we consider the beginning of the De-Rham complex (cf. [FS], Th. 4.1) aug-
mented by its 0th cohomology
0 -! Sd(1)-! Spd -ffi!Spd-1 1
and we put sd(1)to be the kernel of the pd-transformation corresponding
to the De-Rham differential ffi. A point which may be overlooked is that
10
sd(1)(V pd) = Sd(1)(V ) regarded as a graded space has degrees of nontrivial
components multiplied by p. Taking into account this phenomenon it is
convenient to say that the Frobenius twist regarded as a functor on the graded
spaces multiplies grading by p (ie. we put Vp(1)i:= Vi and 0 elsewhere).
4 The main theorems
We start with introducing some notation. Let Ai = Ext *(I(i), I(i)), Bi =
(Ai) d k[ d] with a grading in Ai coming from the grading on Ext-groups
and the group algebra placed in degree 0. We endow Bi with a structure
of d-bimodule given by the formula oe.a1 . . .ad efi.~ = aff(1) . . .
aff(d) efffi~. Sometimes it will be more convenient to look at Bias a bimodule
with the action: oe.a1 . . .ad efi.~ = a~-1(1) . . .a~-1(d) efffi~. An
isomorphism between these two structures is given by the map a1 . . .ad
efi7! afi-1(1) . . .afi-1(d) efi. The main computational result of [FS] was
determination of Ai. It is a graded space which is one-dimensional in even
degrees smaller than 2pi and trivial elsewhere. Now it follows easily from the
Exponential Formula that
Ext*(Id(i), Id(i)) = Bi,
as a graded d-bimodule. It is also easy to see that all computations of
([FFSS], sect. V) may be written in the form
Ext *(F (i), G(i)) = (gin(fpr#(Bi)),
(ie. we first apply fpr# to Bi as a left d-module and then we apply gin
to the resulting right d-module). The main result of this paragraph is
determination of a class of functors for which such a description holds.
We start with a slight generalization of the results of [FFSS]
Fact 4.1
1. For any diagrams ~, ~0, ~, ~0 of weight d we have
Ext*(D~(i), S~(i)) = s~(s~(Bi)) = s~(s~(Bi)),
where we apply s~ to the left d-structure and s~ to the right one.
11
0
2. Moreover, for any transformation _ : S~ -! S~ the induced map
0(i)
_(i)*: Ext*(D~(i), S~(i)) -! Ext*(D~(i), S~ )
under the above isomorphisms, may be described in two ways: either
as _(s~(Bi)) or as s~(_(Bi)). Similarly, for any transformation OE :
0 (i)* # ~
D~ -! D~ the induced map OE may be described either as OE (s (Bi))
or as s~(OE# (Bi)).
Proof: The second description in the first part of the fact for ~ = ~ = (1d)
is just ([FFSS], Th. 4.5). The general case follows from the Exponential
Formula. The first description is the Kuhn dual of the second.
We now turn to the proof of the second part of Fact 4.1 To get the first
description we lift _ to some e_: Id -! Id (the existence of such a lift follows
from the projectivity of Id) and consider the commutative diagram
e_(i)* * ~(i) d(i)
Ext*(D~(i), Id(i))-! Ext (D , I )
# m(i)~* # m(i)~0*
_(i)* * ~(i) ~0(i)
Ext*(D~(i), S~(i))-! Ext (D , S ).
We recall from ([FFSS], sect. V), that the vertical arrows are epimorphic and,
according to the first part of the fact, they may be identified respectively
with m~(s~(Bi)) and m~0(s~(Bi)). Moreover, since _eis just multiplication
by an element of k[ d], we have _e(i)*= _e(s~(Bi)). Hence if we replace
_(i)*by _(s~(Bi)), the diagram remains commutative. But since the left
vertical arrow is onto, there is at most one bottom arrow making the diagram
commutative. Thus _(i)*= _(s~(Bi)).
In order to obtain the second description we consider the diagram
_(i)* * d(i) ~0(i)
Ext *(Id(i), S~(i))-! Ext (I , S )
# j(i)* # j(i)*
_(i)* * ~(i) ~0(i)
Ext*(D~(i), S~(i))-! Ext (D , S ),
where j~ : D~ -! Id is the natural inclusion. After identifying known arrows
we get
_(Bi) * d(i) ~0(i)
Ext*(Id(i), S~(i))-! Ext (I , S )
0
# j#~(s~(Bi)) # j#~(s~ (Bi))
_(i)* * ~(i) ~0(i)
Ext *(D~(i), S~(i))-! Ext (D , S ).
12
(for the vertical arrows we use a description which is Kuhn dual to that from
[FFSS] while for the top arrow we use the previous description for ~ = (1d)).
By the epimorphicity of the left vertical arrow it suffices to observe that
s~(_(Bi)) makes the diagram commutative. The case of a transformation
between divided powers follows from the Kuhn duality. ||
As it was seen in the proof, all the assertions of Fact 4.1 were quite formal
consequences of ([FFSS], Th. 4.5) where the groups Ext *(Dd(i), Sd(i)) were
computed. But this generalization, technically rather straightforward, will
turn out to be extremely useful, for {D~} (resp. {S~}) form the set of projec-
tive (resp. injective) generators of P d. Therefore our strategy for computing
Ext-groups will be, roughly speaking, as follows. To compute Ext*(F (i), G(i))
we take a resolution of F by (sums of products of) divided powers and a res-
olution of G by symmetric powers, we twist them i times and we compute
Ext-groups between the (twisted) resolutions. By Fact 4.1 we know these
Ext-groups and also the arrows between them. This, under some additional
hypotheses, will enable us to calculate the original Ext-groups.
For some technical reasons we will also need an ä dditive analogueö f
the last fact. Put A0j= Hom (jI, I), B0j= (A0j) d k[ d], where jI denotes
I j. Thus, A0jis just j-dimensional space concentrated in degree 0.
Fact 4.2
1. For any diagrams ~, ~0, ~, ~0 of weight d we have
Ext *(D~ O jI, S~) = s~(s~(B0j)) = s~(s~(B0j)),
where we apply s~ to the left d-structure and s~ to the right one.
0
2. Moreover, for any transformation _ : S~ -! S~ the induced map
0
_* : Ext*(D~ O jI, S~) -! Ext*(D~ O jI, S~ )
under the above isomorphisms, may be described in two ways: either
as _(s~(B0j)) or as s~(_(B0j)). Similarly, for any transformation OE :
0 *
D~ -! D~ the induced map (OE O jI) may be described either as
OE# (s~(B0j)) or as s~(OE# (B0j)).
Proof: First, observe that by the projectivity of D~ O jI and the injectivity
of S~ the map s~(s~(B0j)) -! Hom (D~ OjI, S~) is an epimorphism. Hence, it
13
suffices to show that both the spaces have the same dimensions. According
to the Exponential Formula it suffices to do this for ~ = ~ = (1d). Then
sd(sd(B0j)) = Sd(A0j), while
L i i i i
Hom (Dd O jI), Sd) = i1+...+ij=dHom(D 1, S 1) . . .Hom (D j, S j) =
L i 0 i 0
= i1+...+ij=dS 1(A1) . . .S j(A1).
The dimensions of these spaces are clearly equal.
The proof of the second part goes in a similar fashion to that of the
second part of Fact 4.1. The only difference is that the epimorphicity of
vertical arrows in the diagram
e_* * ~ d
Ext *(D~ O jI, Id) -! Ext (D O jI, I )
# m~* # m~0*
_* * ~ ~0
Ext *(D~ O jI, S~) -! Ext (D O jI, S )
immediately follows from the projectivity of D~ O jI. This concludes the
proof of Fact 4.2. ||
We are now in a position to state our first main result.
Theorem 4.3
1. For any F 2 P dand any diagram ~ of weight d:
Ext*(D~(i), F (i)) = fin(s~(Bi)),
where fin is an arbitrary injective symmetrization of F .
0 (i)*
2. For any transformation OE : D~ -! D~ , the induced map (OE ) :
0(i) (i) * ~(i) (i)
Ext *(D~ , F ) -! Ext (D , F ), under the above isomorphisms
takes the form fin(OE# (Bi)).
Also ä dditive analoguesö f these formulae hold, ie.
Hom (D~ O jI, F ) = fin(s~(B0j)),
and (OE O jI)* = fin(OE# (B0j)).
14
Proof: We start by proving the additive version of the theorem. In order to
get
Hom (D~ O jI, F ) = fin(s~(B0j)),
0 d
we extend the map _0(V d) : fin(V d) -! s~ (V ) to a resolution
_0 ~0 _1 ~1 _2 _l ~l
0 -! F - ! S -! S -! . .-.! S -! 0,
(from now on we will slightly abuse notation denoting by the same letter a
d-transformation and its evaluation on V d). Since Ext n(D~ O jI, F ) = 0
for n > 0, this complex remains exact after applying Hom (D~ O jI, -). Let
us consider the diagram
_0(s~(B0j))0 _1(s~(B0j))1 _2(s~(B0j))
0 -! fin(s~(B0j)) -! s~ (s~(B0j)) -! s~ (s~(B0j)) -! . . .
# #
_0* ~ ~0 _1* ~ ~1 _2*
0 -! Hom (D~ O jI, F ) -! Hom (D O jI, S )- ! Hom (D O jI, S ) -! . . .
whose bottom row is exact. Note moreover, that thanks to Fact 4.2, the
vertical arrows (exist and) are isomorphisms and that the top row is exact by
Fact 3.3.2 (s~(B0j) is a Y -permutative module because it is a tensor product
of two Y -permutative modules). Thus, we get an isomorphism fin(s~(B0j)) '
Hom (D~ OjI, F ) by an easy diagram chasing. Let us notice for the future use
that under this identification we have (_0)* = _0(s~(B0j)). This is important
because this time it need not to be true that _0(s~(B0j)) = s~(_0(B0j)). The
easiest example of this pathology is provided by the arrow Hom (Dp, Dp) -!
Hom (Dp, Ip) induced by the inclusion _0 : Dp -! Ip which may be thought
of as the beginning of an injective resolution of Dp. Indeed: in this case
sp(_0(B01)) is trivial. The existence of such phenomena will make us to be
very careful in the further arguing.
We now turn to the second part of the additive version of the theorem. A
0
transformation OE : D~ - ! D~ induces a commutative diagram with exact
rows
0 0 ~0 0
0 0 _0(s~ (Bj))~0 ~0 0 _1(s (Bj))~1 ~0 0
0 -! fin(s~ (Bj)) - ! s (s (Bj)) - ! s (s (Bj))
0 # 0 ~1 # 0
# OE(jI)* # s~ (OE (Bj)) # s (OE (Bj))
_0(s~(B0j))0 _1(s~(B0j))1
0 -! fin(s~(B0j)) -! s~ (s~(B0j)) -! s~ (s~(B0j)).
15
Now it suffices to observe that replacing OE(jI)* by fin(OE# (B0j)) does not de-
stroy the commutativity of the diagram. It means that OE(jI)* = fin(OE# (B0j)).
Again it turns out that we could not take OE# (fin(B0j)) instead of fin(OE# (B0*
*j)).
A simple example of the arrow Hom (Ip, Dp) -! Hom (Dp, Dp) induced by the
inclusion OE : Dp -! Ip shows, that the maps fin(OE# (B0j)) and OE# (fin(B0j))
need not to coincide. It looks strange because, as we remember from Fact 4.2,
in all further vertical arrows they do coincide. But I recall that we cannot
change the order of applying d-functors also on the left horizontal arrows.
We now turn to the proper version of Theorem 4.3. This time we first twist
i times an injective resolution of F and then we apply to it Ext *(D~(i), -).
According to Fact 4.1 we get the commutative diagram
_0(s~(Bi))~0 ~ _1(s~(Bi))~1 ~ _2(s~(Bi))
0 -! fin(s~(Bi)) -! s (s (Bi)) -! s (s (Bi)) -! . . .
# #
(_(i)0)** ~(i) ~0(i)(_(i)1)* * ~(i) ~1(i) (_(i)2)*
0 -! Ext*(D~(i), F (i)) - ! Ext (D , S ) -! Ext (D , S ) - ! . .*
* .
in which all vertical arrows are isomorphisms. In order to finish the proof like
in the additive version it is sufficient to show that the bottom row is exact.
But we know that it is exact at least starting from the third term, because
the top row is exact. It means that in the first hiperExt spectral sequence
converging to hExt *(D~(i), C) = 0 (where C stands for the twisted resolution
of F ), the E2-term may be nontrivial only at the first two columns. There-
fore, by dimension argument, it must be trivial (this argument generalizes
(and may be easily derived from) a well known fact that if every third arrow
in a long exact sequence is epimorphic then the sequence splits).
The proof of the second part is analogous to the proof of the additive
counterpart. This completes the proof of Theorem 4.3 ||
In order to repeat the argument with respect to the second variable we will
need an assumption guaranteeing exactness of a complex in the situation
when Fact 3.3.2 is not applicable. Moreover, problems with functoriality
make the formulation of the result more complicated and make us to intro-
_0 ~0
duce another bit of notation. Let 0 - ! G - ! S be the beginning of
0 ffi0
an injective resolution of G and D~ - ! F -! 0 be the beginning of a
16
projective resolution of F . We consider the commutative diagram
(_(i)0)* * (i) ~0
Ext *(F (i), G(i))-! Ext (F , S )
# (OE(i)0)* # (OE(i)0)*
0 (i) (_(i)0)* * ~0(i) ~0(i)
Ext *(D~ , G ) - ! Ext (D , S ).
According to Theorem 4.3 we may rewrite it as
(_(i)0)* pr# ~0
Ext*(F (i), G(i)) -! f (s (Bi))
0
# (OE(i)0)* # OE#0(s~ (Bi))
0 _0(s~0(Bi))~0 ~0 ~0 ~0
gin(s~ (Bi)) -! s (s (Bi)) = s (s (Bi)).
0 # ~0
We put (fpr#, gin)(Bi) to be im(_0(s~ (Bi))) \ im (OE0 (s (Bi))). The point
of this definition is that in general we cannot identify this space neither with
fpr#(gin(Bi)) nor with gin(fpr#(Bi)). Nevertheless, this is certain explicitly
defined space which is determined by the symmetrizations fin, gpr. Quite
naturally, this space will be our candidate for Ext*(F (i), G(i)) in general.
Theorem 4.4 Assume that Ext*(F O piI, G) = 0 for * > 0 (we will call this
assumption the "Ext-condition"). Then
Ext *(F (i), G(i)) = (fpr#, gin)(Bi).
Proof: We take an injective resolution of F , a projective resolution of G and
consider the diagram
0 0 0
# # #
(_(i)0)* * (i) ~0(i) (_(i)1)* * (i) ~1(i)
0 -! Ext*(F (i), G(i)) - ! Ext (F , S ) - ! Ext (F , S ) -!
# (OE(i)0)* # (OE(i)0)* # (OE(i)0*
*)*
0(i) (i) (_(i)0)* * ~0(i) ~0(i) (_(i)1)* * ~0(i) ~1(i)
0 -! Ext*(D~ , G ) - ! Ext (D , S ) - ! Ext (D , S ) -!
# (OE(i)1)* # (OE(i)1)* # (OE(i)1*
*)*
1(i) (i) (_(i)0)* * ~1(i) ~0(i) (_(i)1)* * ~1(i) ~1(i)
0 -! Ext*(D~ , G ) - ! Ext (D , S ) - ! Ext (D , S ) -!
# # #
17
According to Theorem 4.3 all the rows except perhaps the first and all the
columns except perhaps the first are exact. Hence, the prof will be finished
by a diagram chasing if we show that the first column is exact. By Theorem
4.3 we have in the first column the sequence
(ffi(i)0)*in~0 ffi#1(gin(Bi))in~1 ffi#2(gin(Bi))
0 -! Ext*(F (i), G(i)) - ! g (s (Bi)) - ! g (s (Bi)) - ! . (*)
Now we consider the sequence
ffi2 ~1 i ffi1(piI)~0 i ffi0(piI) i
. .-.! D O p I -! D O p I -! F O p -! 0.
Since it is a projective resolution of F O piI and Ext n(F O piI, G) = 0 for
n > 0, the sequence
(ffi0(piI))* ~0 i (ffi#1(piI))* ~1 i (ffi#2*
*(piI))*
0 -! Hom *(F OpiI, G) -! Hom (D Op I, G) -! Hom (D Op I, G) -!
is exact. But thanks to Theorem 4.3 we may rewrite it as
# in 0 # in 0
(ffi0(piI))*in~0 0 ffi1(g (Bpi))in~1 0 ffi2(g (Bp*
*i))
0 -! Hom *(F OpiI, G) -! g (s (Bpi)) -! g (s (Bpi)) -! .
We now observe that if we neglect the grading then, starting from the second
term, our sequence as a sequence of vector spaces is isomorphic to (*), be-
cause dim(Bi) =dim(B0pi). Therefore our first column is exact starting from
the third term (in order to use this argument we have introduced all these
ä dditive analogues"). Thus by the hiperExt-argument which we used at
the end of the proof of Theorem 4.3, the whole column must be exact. This
completes the proof of Theorem 4.4. ||
5 Applications to classical functors
We would like to find functors satisfying the assumption of Theorem 4.4. The
most important example is provided by F = W~=~0and G = S~=~0. Indeed,
Fact 2.1 together with the Decomposition Formula show that they satisfy
the Ext-condition. Moreover, in this particular case the statement of the
theorem may be formulated in a much simpler way.
18
Theorem 5.1 For any skew diagrams ~=~0, ~=~0 of weight d we have
Ext*(W~(i)=~0, S(i)~=~0) = s~=~0(s~=~0(Bi)) = s~=~0(s~=~0(Bi)).
Moreover, for any transformation _ : s~=~0- ! s~1=~10the induced map _* :
Ext*(W~(i)=~0, S(i)~=~0) - ! Ext *(W~(i)=~0, S(i)~1=~10) takes the form _(s~=~0*
*(Bi)) =
s~=~0(_(Bi)). An analogous fact also holds for transformations of the first
variable.
Proof: When we look once again at the proof of Theorem 4.4, we see that
the reason for which we could not obtain a simpler description of the Ext-
0 * (i)
groups was that in general the map _ : s~ -! s~ induces on Ext (F , -)
the map fin#(_(Bi)) which may be different from the map _(fin#(Bi)). We
will show that for F = W~=~0and G = S~=~0these two maps coincide. By
arguments used in the proof of the second part of Fact 4.1 it suffices to show
the lemma (which is very specific to Weyl and Schur functors):
Lemma 5.2 For any diagrams ~=~0 and ~, the map m~ : id -! s~ induces
an epimorphism Ext *(W~(i)=~0, Id(i)) -! Ext*(W~(i)=~0, S~(i)).
Proof: Of course, it suffices to show the additive version of the lemma. Ap-
plying the Decomposition Formula to W~=~0OpiI, we reduce the proof to show-
ing that there is an epimorphism Hom (W~=~0, Id) -! Hom (W~=~0, S~). To
do this it suffices to show that Ext1(W~=~0, ker(m~)) = 0. By the Littlewood-
Richardson rule, any skew Schur functor has a filtration with a graded object
being a sum of Schur functors. Thus it suffices to establish the last formula
for ~0= ;. Using the Littlewood-Richardson rule again, we observe that the
structural inclusion W~ -! ~ has a cokernel with a graded object being a
sum of Schur functors for diagrams lexicographically smaller than ~ (the rea-
son is that ~ is the largest diagram appearing in the Littlewood-Richardson
decomposition of ~). Thus using induction on the lexicographic order we
reduce our task to showing that Ext 1( ~, ker(m~)) = 0. By the Decompo-
sition Formula, the last statement is equivalent to the fact that m~ induces
an epimorphism Hom ( d, Id) -! Hom ( d, S~), which is clear by Fact 2.1.
This completes the proof of the lemma. ||
Therefore we may choose a more convenient order of applying d-functors
in the main diagram in the proof of Theorem 4.4. In particular, we may
19
identify the morphism (_(i)~=~0)* : Ext *(W~(i)=~0, S(i)~=~0) -! Ext *(W~(i)=~0*
*, S~g=~0(i))
with the map _~=~0(s~=~0(Bi)) = s~=~0(_~=~0(Bi)). This enables us to identify
the Ext-groups with spr#~=~0(sin~=~0(Bi)) or sin~=~0(spr#~=~0(Bi)). In a simila*
*r fashion
we obtain the desired description of induced maps. The second description
of Ext-groups is the Kuhn dual of the first. ||
Thus we have obtained a nice description of Ext-groups between twisted
Weyl and Schur functors. In particular for ~ = (1d) and ~ = (d) we get
respectively
Ext *(Dd(i), S(i)~=~0) = s~=~0(sd(Bi)) = s~=~0(Aid) = S~=~0(Ai),
and
Ext *( d(i), S(i)~=~0) = s~=~0(~dcoinv(Bi)) = s~=~0((Aid)alt) = Wg~=~0(Ai),
strongly generalizing computations of ([FFSS], sect. V). The only computa-
tionin in [FFSS] among those concerning diagrams of the same weight (we
deal with diagrams of different weights in [C2]) which does not fit this scheme
is a computation of Ext *(Dd(i), Dd(i)). But these (and more general) groups
may be computed directly from our Theorem 4.3. Indeed, it yields the for-
mula
Ext*(Dd(i), W~(i)=~0) = w~=~0(sd(Bi)) = w~=~0(Aid) = W~=~0(Ai).
On the other hand, one should be cautious using Theorem 5.1. For example,
it is easy to see that the epimorphism _ : Ip -! p induces the trivial map
Hom ( p, Ip) -! Hom ( p, p) which seems to contradict Theorem 5.1 which
says that it should be an epimorphism. The point is that we should consider
the second variable as a Schur functor so its appropriate symmetrization is
~pinv(we cannot take an arbitrary injective symmetrization in Theorem 5.1).
Thus the corresponding d-transformation _e: id -! ~pinvis the averaging
to invariants which is not an epimorphism in general.
Let us now try to look for other functors satisfying the assumption of The-
orem 4.4. Taking into account Fact 2.2, it is tempting to consider Schur
functors S~, S~ satisfying ~ 6 *~, since we have Ext *(S~, S~) = 0 for * > 0.
But in fact, we need the stronger condition Ext*(S~ O jI, S~) = 0 for j = pi.
When we apply the Decomposition Formula to S~ O jI we see that our lexico-
graphic assumption is quickly weakening. A counterexample is very simple:
20
already for ~ = ~ = (22), p = 2 we get Ext2(S~ O 2I, S~) 6= 0. A pathological
element comes from the decomposition of ~ into (12), (12) and ~ into (2), (2).
Also as small lexicographically diagram as (2k - 1, 1) and as large as (2k)
may be decomposed to give a nontrivial element in Ext*(S(2k-1,1)O 2I, S(2k))
for p = 2. Slightly more complicated examples can be constructed for p > 2,
and even for hooks. The only quite general class of Schur functors satisfying
the assumption of Theorem 4.4 is provided by diagrams of weight p.
Corollary 5.3 If ~ 6*~ are diagrams of weight p then
Ext *(S(i)~, S(i)~) = (w~, s~)(Bi)).
Proof: Observe that when we decompose ~ into smaller diagrams we get
diagrams of weight smaller than p for which Schur functors are projective.
This together with the lexicographic assumption gives the Ext-condition. ||
In the above case there is no reason for expecting that the formula will
simplify to the form similar to that of Theorem 5.1. But the general formula
from Theorem 4.4 is not very convenient in practice. In order to rephrase
the result in a more explicite form, we will show one easy general fact.
Fact 5.4 Let 0 - ! F1 -! . . .-! Fk - ! 0 be an exact sequence whose
all objects satisfy the Ext-condition with some G. Then the sequence: 0 -!
Ext*(Fk(i), G(i)) -! . .-.! Ext*(F1(i), G(i)) -! 0 is exact.
Proof: The assertion follows immediately from the fact that, according to
Theorem 4.4, all Ext-groups under consideration are concentrated in even
degrees. ||
Again it seems that the last result is in conflict with the fact that _ : Ip -!
p induces the trivial map Hom ( p, Ip) -! Hom ( p, p). But the point is
that _ cannot be extended to an exact sequence satisfying the Ext-condition
with Dd.
Now we would like to get a more explicit description of Ext-groups ap-
pearing in Cor. 5.3. To this end, we will need a resolution of S~ by exterior
powers starting with the structural arrow OE~. The existence of such a reso-
lution may be derived from some corollary of Theorem 5.1 and Fact 5.4.
_~ e~
Corollary 5.5 For any resolution 0 -! S~ -! S -! . .,.there exists the
ffi#
öK szul dual complex" 0 -! We~-! e~ e~-! . .,.which is exact.
21
_~
Proof: We apply the functor Ext *( d(i), -) to the complex 0 -! S(i)~-!
ffi#(Ai)
Se~(i)-! . .A.ccording to Theorem 5.1 we get the complex 0 -! We~(Ai) -!e~
e~(Ai) -! . .,.whose exactness follows from Fact 5.4. Since the dimension
of Ai may be arbitrarily large, the whole complex of functors must be exact.
||
To obtain the desired resolution of S~ by exterior powers, we take the
ffi#~~
Kuhn dual of the complex 0 -! W~ -! -! . .c.onstructed in Cor. 5.5
(for ~ := e~). Thus we get the resolution
1 ffi1 ~ ffi~
. .-.! ~ -! - ! S~ -! 0.
Now observe that since exterior powers are also Weyl functors, they satisfy
the Ext-condition with S~. Since S~ satisfy it too, the sequence
(ffi~)* * ~(i) (i) (ffi1)*
0 -! Ext*(S(i)~, S(i)~) -! Ext ( , S~ ) -! . . .
is exact by Fact 5.4. Hence Ext *( ~(i), S(i)~) = ker((OE1)*). In order to com-
pute this kernel let us observe that all groups and arrows appearing in the
above sequence starting from the second term are known by Theorem 5.1:
1 ~
Ext*( ~(i), S(i)~) = ~~i(s~(Bi)) etc. Let eOE1: ~~c -! ~c be a d-functor such
that eOE1(V d) = OE1. The proof of existence of such a d-transformation is
analogous to the proof of epimorphicity in Lemma 3.4. Therefore, when we
put fl := coker(OeE1), we get
Ext *(S(i)~, S(i)~) = fl# (s~(Bi)).
Thus we obtained a description of the Exts in terms similar to those used
in Theorem 5.1. One should remember however, that although fl is quite
explicitly defined symmetrization of W~, one cannot expect that fl ' w~
and even that fl# (s~(Bi)) ' w~(s~(Bi)), for s~(Bi) is not a Y -permutative
module.
Finally, I would like to discuss some interesting special cases of Theorem 4.3.
When we take ~ = (d), the formula takes the nice form
Ext *(Dd(i), F (i)) = fin(sd(Bi)) = fin(Aid) = F (Ai).
22
This is the first time when we obtained the description clearly independent
of the choice of fin. Let us look at a very simple but instructive example.
When we take F = I(i), then our formula gives Ext *(Dp(i-1), I(i)) = A(1)i-1.
We recall however, that the decoration (1)indicates that we should multiply
by p degrees of nontrivial components of Ai-1. After this modification we
get the result predicted by ([FS], Th. 4.5).
The second extreme case: ~ = (1d) is even more interesting. We get
Ext *(Id(i), F (i)) = fin(Bi).
But these groups can also be computed directly form the Decomposition
Formula. Namely, let us put gcrd(F )(V ) := crd(F )(V, . .,.V ), where crd(F )
means the d-th cross-effect of a functor F (see eg. [B1], pp. 74-75). Then,
sinceLF has the Eilenberg-MacLane degree at most d, we have crd(F )(V1, . .,.Vd*
*) =
(V1 . . .Vd). Therefore:
Ext*P(Id(i), F (i)) = Ext*Pd(V1(i) . . .Vd(i), F (i)(V1 . . .Vd)) =
Ext*Pd(V1(i) . . .Vd(i), crd(F )(i)(V1, . .,.Vd)) =
L (i) (i)
Ext*Pd(V1(i) . . .Vd(i), (V1 . . .Vd )) =
L * (i) (i) * (i) (i)
= (Ext P (I , I ) . . .ExtP (I , I )) = gcrd(F )(Ai),
and it is easy to see that a d-structure on Ext *(Id(i), F (i)) agrees with a
structure on gcrd(F )(Ai) coming from the permutation of factors in crd(F ).
Thus we get the formula
fin(Bi) = gcrd(F )(Ai),
which suggests the existence of a link between the notion of symmetrization
and cross-effects.
Let us examine more closely the case i = 0. In this case the space A0 is
one-dimensional, hence our computation simplifies to the formula
fin(k[ d]) = Hom Pd(Id, F ).
Regarding the right-hand side of this equality as a functor of the second
variable we get the functor S : Pd - ! k[ d]-mod . This functor known
(unfortunately) also as the Schur functor is an important tool used to com-
pare representations of the general group and symmetric group (see eg. [Ma],
23
chap. 4). The last equality may serve as an "explanationö f some good
properties of the Schur functor. Namely, our formula says that the functor
F and the d-module S(F ) are obtained by applying the same d-functor
to the right d-structure of: V d in the first case, and k[ d] in the second.
Acknowledgements. This article (together with [C1] and [C2]) contains re-
sults of my Ph.D. thesis, written under the supervision of Stanis_law Betley
at the University of Warsaw. I wish to express my gratitude to professor
Betley for many valuable suggestions and strong encouragement.
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25