Extensions of Weyl and Schur functors
Marcin Cha_lupnik *
Instytut Matematyki, University of Warsaw
ul. Banacha 2, 02-097 Warsaw, Poland
e-mail: mchal@mimuw.edu.pl
Abstract
We study Ext-groups between twisted Weyl and Schur functors in
the category of strict polynomial functors. We give complete descrip-
tion of the groups Ext*P(W~(i+j), S(i)Fj) for |~| = |~| and we obtain
k(~)
some partial results in the general case.
1 Introduction
This paper is a third part of my work on homological algebra in the category
of functors. We continue here the calculations of Ext-groups started in [C1],
armed with the Schur-De-Rham complex studied in [C2].
The main result of the present article (Th. 3.4) is the calculation of the
groups
Ext*P(W~(i+j), S(i)Fj),
k(~)
for diagrams ~, ~ of equal weights (Fkjmeans the j-fold iteration of the
operation Fk introduced in [C2]). This completes the process of generalizing
results of [FFSS] but also opens new prospects for computing Ext-groups
in the functor category. The main challenge is to understand the groups
______________________________
*The author was partially supported by Bia_lynicki-Birula Subsydium of Found*
*ation of
Polish Science.
1
Ext*P(W~(*), S(*)~) for arbitrary Young diagrams ~, ~. Section 4 contains some
partial results related to this problem, which suggest further connections of
the subject with the combinatorics of hooks.
2 Acyclic functors and localization
Throughout this article we work in the category P of strict polynomial func-
tors over a fixed field k of characteristic p > 0 (cf. [FS, Sect. 2]). We will
use freely all definitions and conventions made in [C1, Sect. 2,3] and [C2,
Sect. 2,3,5,6]. In particular, we recall that ~ may denote a skew Young di-
agram. All other references to [C1] and [C2] (on which the present article
depends heavily) will be explicitly mentioned.
Our main goal is generalization of [C1,Th. 5.1] to the case of diagrams of dif-
ferent weights. Taking into account the role played by the De-Rham complex
in the proof of [FFSS, Th. 4.5] and the fact that we computed the cohomol-
ogy of the Schur-De-Rham complex for diagrams of the form Fk(~) in [C2,
Th. 5.3], we can expect that the groups Ext*P(W~(i+j), S(i)Fj) admit descrip-
k(~)
tion similar to that of [C1, Th. 5.1]. One should remember however, that the
pair W~(j), SFjk(~)does not satisfy the "Ext-conditionö f [C1, Th. 4.4]. Also,
when we try to repeat the proof of [C1, Th. 4.4], we face the problem of con-
j(~) Fj(~)
structing maps SFjk(~)-! SFk or k - ! SFjk(~)which could serve as
analogues of structural arrows. However lexicographic properties of diagrams
do not guarantee us the existence of such maps any more, we will manage
to perform the calculations in a fashion similar to that of [C1]. The point is
that we need not maps between functors but only between their Ext-groups.
But the Schur functors associated to diagrams which form combinatorial ob-
stacles for constructing the analogues of structural maps will turn out to
have trivial Ext-groups. The situation resembles that in [C2] where only
after neglecting some acyclic complexes we were able to construct maps on
cohomology (in fact, as we will see, the combinatorics in both contexts is
similar). In order to make considerations of this sort precise and functorial
we will use a formalism of localization of the derived category.
Definition 2.1 Given A 2 P, we say that a triangulated category DP A is
an A-localization of the derived category DP if there exists a functor L :
P -! DP A such that
2
o L takes short exact sequences to distinguished triangles.
o For all X 2 P,
ExtnP(A, X) = Hom DP A(L(A)[n], L(X)).
o If f : X -! Y induces an isomorphism f* : Ext*P(A, X) ' Ext*P(A, Y )
then L(f) is an isomorphism.
The existence of such a localization is well known (see eg. [Ne, Chap. 2.1]).
The second condition allows us to carry over all the calculations to the local-
ized category, hence from now on we will not differ between Hom DP A(L(A)[n], *
*L(X))
and Ext nP(A, X) usually denoting both just by Ext n(A, X). The most im-
portant is the third condition which makes constructing maps much easier.
We start with quite formal lemma which enables us to formulate further
results in a concise manner.
Lemma 2.2 For any diagram ~ of weight d, we have the following isomor-
phisms in the category DP W(i+j)~:
j(i)j
S(i)Fj' dp [hk(~)],
k(~)
for an arbitrary diagram ~ of weight d consisting of a single row, and
j(i) j 0
S(i)Fj '0Sdp [hk(~ )],
k(~ )
for an arbitrary diagram ~0 of weight d consistingjof a single column.
The shift is given by the formula hjk(ff) = k p_-1_p-1+ (pj- 1)fff, where fffst*
*ands
for the number of boxes lying above the principal diagonal.
Proof: Since the proofs for ~ and ~0 are analogous, we focus on the case of
one-rowed diagram. The idea of proof is similar to that of [C2, Lemma 7.1],
the difference is that this time we deal with pj-hooks instead of p-hooks.
The proof goes by induction on the number of rows in ff = Fkj(~). We prove
the assertion for a slightly wider class of diagrams, namely for skew hooks
ff whose "pj-slices" (definition of pj-slices is analogous to that of p-slices
given in [C2, Sect. 5]; see also discussion after Fact 2.4 in the present artic*
*le)
are placed horizontally (ie. the foot of the next slice lies to the right of the
3
hand of the previous one). In order to get the assertion for a diagram ff, we
consider the exact sequence
0 -! S(i)ff=(ff1)|h(ff1)-! S(i)ff=(ff1) ff1(i)-! S(i)ff-! 0,
whose exactness may be proved by the argument used in [C2, Lemma 7.1].
Since the pj-slices of ff are placed horizontally, the number ff1 cannot be div*
*is-
ible by pj. Hence, by the Decomposition Spectral Sequence (cf. [C1, Sect. 2]),
we get Ext *(W~(i+j), S(i)ff=(ff1) ff1(i)) = 0. this leads to an isomorphi*
*sm
S(i)ff=(ff1)|h(ff1)' S(i)ff[-1] in the localized derived category. But ff=(ff1*
*)|h(ff1)
has less rows than ff, so we can apply the induction hypothesis. The final
j-1 pj-1-k pj*
*-1_
formula for shift comes from the fact that Fkj((1)) = (k p___p-1+ 1, 1 p*
*-1).
||
Now, we focus on the problem of finding functors which are trivial in the
localized category.
Lemma 2.3 Assume that for some A 2 P, s, i, j satisfying s > i + j, and
for some diagram ~ of weight d, we have Ext *(A(s), S(i)Fj) 6= 0. Then ~ has
k(~)
a trivial p-core.
Proof: The proof falls naturally into three parts.
1: If the assertion holds for A1 and A2, then it does so for A1 A2.
Assume that Ext *(A(s)1 A(s)2, S(i)Fj) 6= 0. By the Decomposition Spec-
k(~)
tral Sequence, there must exist ff Fkj(~) such that Ext *(A(s)1, S(i)ff) and
Ext*(A(s)2, S(i)Fj ) are nontrivial. Hence, according to the assertion for A2
k(~)=ff
(and j = 0), Fkj(~)=ff has a trivial core. Then, by [C2, Fact 6.1], there exists
ff0 Fkj-1(~) such that ff = Fk(ff0). Using our assertion again, this time
for j = 1, we conclude that Fkj-1(~)=ff0 has a trivial core, hence there exists
ff00 Fkj-2(~) such that ff0 = Fk(ff00) etc. till we get that ff = Fkj(ffj) and
ffj, ~=ffj have trivial cores. Therefore, also ~ has a trivial core.
2: A = Dd(t).
We proceed by induction on d. Let d = 1 and assume that c(~) 6= ;. Since
we will consider both the classical Schur complex and the Schur-De-Rham
complex introduced in [C2] we will call the former one the Schur-Koszul
complex or we will say that we consider the Schur complex equipped with
4
the Koszul or De-Rham differential (but we denote both the complexes by
S~). We start with taking the twisted Schur-De-Rham complex S(i+j)~. Since
by ([C2], Fact 4.3) it is acyclic and Ext *(I(s+t), B C) = 0 for all homo-
geneous functors of positive degree (eg. by the Exponential Formula [C1,
Sect. 2]), the first spectral sequence converging to hExt *(I(s+t), S(i+j)~) (by
this we mean the hyperExt groups of I(s+t)with coefficients in S(i+j)~(see eg.
[CE, Chap. XVII])) converges to 0 and may have nontrivial the first or last
column only. Thus we obtain Ext*(I(s+t), S(i+j)~) = Ext*+|~|-1(I(s+t), We(i+j)~*
*).
Applying the same argument to the Schur-Koszul complex we get shift
into another direction Ext *(I(s+t), S(i+j)~) = Ext *-|~|+1(I(s+t), We(i+j)~). *
* But
by the hypothesis s > i + j we know that |~| > 1. This means that
Ext*(I(s+t), S(i+j)~) = 0 = Ext*(I(s+t), We(i+j)~).
Now, we show in a similar manner that Ext *(I(s+t), S(i+j-1)Fk(~)) = 0. This
time the Schur-De-Rham complex S(i+j-1)Fk(~)is not acyclic, but since by [C2,
Th. 5.3] the second spectral sequence converging to hExt *(I(s+t), S(i+j-1)Fk(~*
*))
is trivial, we still have the shift in grading between Ext *(I(s), S(i+j-1)Fk(~*
*)) and
Ext*(I(s), WF(i+j-1)), which gives the desired vanishing of the Ext-groups. Re-
k(e~)
peating this argument we get Ext *(I(s+t), S(i+j-q)Fq) = 0 for larger and larger
k(~)
q. At last for q = j we obtain our assertion.
The proof of the induction step on d is similar. We assume the assertion
for all d < d0. In order to get it for d0 we look at the spectral sequences
converging to hExt *( d0(s+t), S(i+j)~) (equipped with the De-Rham differen-
tial). By the induction hypothesis and part 1, the second spectral sequence
is trivial. Since for the same reason the first sequence has at most two non-
trivial columns, we get the shift in grading. The shift into another direction
is provided by the Koszul complex.
3: The general case.
For arbitrary A we take a resolution by products of divided powers. By parts
1 and 2, the assertion holds for all functors in the resolution. Therefore it
also holds for A. ||
Remark: Observe that even for s = 1, i = j = 0, Lemma 2.3 is not obvious.
For solid ~ it follows from the fact that each twisted functor belongs to the
trivial block and the Nakayama Conjecture for P [Do]. But this argument
5
fails for skew diagrams.
We use this lemma to derive a powerful criterion for detecting functors
with trivial Ext-groups.
Fact 2.4 If for some fi Fkj(~) and some A 2 P, Ext *(A(i+j), S(i)Fj ) is
k(~)=fi
nontrivial, then fi = Fkj(ff) for some ff ~.
Proof: Assume that the above Ext -group is nontrivial . Then, by Lemma
2.3 (for j = 0), c(fi) = ;. Hence, by [C2, Fact 6.1] we get fi = Fk(fi0).
Iterating this argument we obtain our assertion (we have used a similar trick
in the proof of part 1 of Lemma 2.3). ||
It seems that we have been doing the same work many times in the last
proofs. Let us look more closely at the situation from the point of view of
combinatorics. Since we are interested in multi-twisted functors, we should
consider (in contrast to [C2]) the operation Fkjof j-fold enlargement of a
diagram. It is completely analogous to Fk, the only difference is that we
replace boxes by pj-hooks instead of p-hooks. Namely, to build Fkj(~) out of
~, we replace the boxes in ~ lying above the diagonal by the horizontal pj-
hooks, those below it by the vertical ones, and we replace the boxes lying on
j-1 pj-1-k pj-1_
the diagonal by the hooks of shape Fkj((1)) = (k p___p-1+ 1, 1 p-1). Thus
it would be tempting to derive Fact 2.2 from the "pj-analogueö f [C2, Fact
6.1] which obviously holds. Unfortunately it not true that if Ext *(A(j), S~)
is nontrivial, then ~ has a trivial pj-core (we will come back to this problem
in Section 4).
j(~) Fj(e~)
Now, we shall construct the arrows Fk -! SFjk(~), SFjk(~)-! S k
which could play roles of the structural arrows for enlarged diagrams. It will
turn out that the construction which fails in the category P is possible in
the localized category. To see this, let us investigate the combinatorics of
j(~)
the situation. An attempt to construct an arrow Fk - ! SFjk(~)in the
category P breaks down because the diagram Fkj((~1, . .,.~l-1)) is not the
lexicographically smallest subdiagram of a given weight in Fkj((~1, . .,.~l))
etc. We will show however, that all smaller diagrams give trivial objects in
DP W(i+j)~. To see this, let us take fi Fkj(~) such that Ext *(W~(i+j), S(i)*
*fi
S(i)Fj ) 6= 0. Then, by the Decomposition Spectral Sequence, there exists
k(~)=fi
6
fl ~ of weight |fi|=pj such that Ext*(Wf(i+j)l, S(i)fi) Ext*(W~(i+j)=fl, SF*
*jk(~)=fi) is
nontrivial. Hence, by Fact 2.4, fi = Fkj(ff). But among diagrams of the form
Fkj(ff0), our diagram is the smallest (of a given weight). This observation
(note that the underlying combinatorics is completely analogous to that of
the öH mological Decomposition Formula" in [C2, Sect. 6]) enables us to
construct the arrow
j(~)(i) (i)
Fkj(i)(OE~) : Fk -! SFj
k(~)
in the category DP W(i+j)~, and by a similar reasoning, the map
j(e~)(i)
Fkj(i)(_~) : S(i)Fj-! SFk .
k(~)
Next, observe an interesting fact, that the composition Fkj(i)(_~) O Fkj(i)(OE~)
exists already in P, for it is equal to the composition of the öc multiplicatio*
*n"
j(~)(i) Fj(~)(i) Fj(e~)(i)
and üm ltiplication": Fk -! I k -! S k . In the last formula
j(~)(i)
IFk stands for the tensor product of twisted Schur functors corresponding
to the pj-slices in Fkj(~) (we recall the interpretation of the operation Fkj
in terms of pj-hooks given after the proof of Fact 2.4). The first arrow
is the tensor product of Fkj(i)(OE(~s)) for all rows of ~, while the second is
the product of Fkj(i)(_(e~s)) for all columns of ~. It is easy to see (and we
have taken advantage of this in [C2, Sect. 7]) that for one-rowed (or one-
columned) diagrams the combinatorial obstacles for the existence of maps in
P disappear.
Similarly, in a dual situation we define the "structural maps": Fkj(i)(OE#~)*
* :
j(~)(i) j(i)# Fj(~)(i)
WF(i)j- ! Fk , Fk (_~ ) : D k - ! W j e , whose composition
k(~) Fk(~)(i)
exists in P. Moreover, thanks to Lemma 2.2 and [FFSS, Th. 4.5], we are
able to describe the maps induced on Ext-groups by these compositions. In
order to express them concisely, we shall introduce notation analogous to
j(i)
that of [C1, Sect. 4]. Let Aij= Ext*(I(i+j), Sp ) (this is a one-dimensional
space in degrees divided by 2pj, less than 2pi+j, and trivial elsewhere [FS,
Th. 4.5]), and let Bij = Aidj k[ d]. The space Bij is endowed with a
structure of a d-bimodule defined by the formula known from [C1, Sect. 4]:
oe.a1 . . .ad efi.~ = aff(1) . . .aff(d) efffi~. Slightly generalizing the
results of [FFSS, Sect. 4, 5] by means of the Exponential Formula and Lemma
2.2, we obtain
7
Fact 2.5 The following isomorphisms hold:
j(~)(i) ~ d j ~ j
Ext *(Dd(i+j), Fk ) = ~inv(s (Bij))[hk(~)] = (Aij)[hk(~)],
j(~)(i) ~ d j ~ j
Ext*(Dd(i+j), SFk ) = s (s (Bij))[hk(~)] = S (Aij)[hk(~)],
j(~)(i) ~ d 0j ~ 0j
Ext *(Dd(i+j), DFk ) = d (s (Bij))[hk(~)] = D (Aij)[hk(~)],
j(~)(i) ~ d j ~ j
Ext *( d(i+j), Fk ) = ~inv(~inv(Bij))[hk(~)] = D (Aij)[hk(~)],
j(~)(i) ~ d j ~ j
Ext *( d(i+j), SFk ) = s (~inv(Bij))[hk(~)] = (Aij)[hk(~)],
the shifts are given by the formulae:
pj - 1
hjk(~) = f~(pj - 1) + e~k ______,
p - 1
pj - 1
h0jk(~) = (2d - f~)(pj - 1) - e~k ______,
p - 1
where e~, f~ denote respectively the number of boxes lying on and above the
main diagonal.
Under these identifications the map induced on Ext*(Dd(i+j), -) by the
map Fkj(i)(_~)OFkj(i)(OE~) is equal to _~OOE~(sd(Bij))[hjk(~)] = _~OOE~(Aij)[hj*
*k(~)],
and the one induced by Fkj(i)(OE#~)OFkj(i)(_#~) is equal to OE#~O_#~(sd(Bij))[h*
*0jk(~)] =
OE#~O _#~(Aij)[h0jk(~)].
Similarly, Fkj(i)(_~) O Fkj(i)(OE~) induces on Ext*( d(i+j), -) the map OE#~O
_#~(sd(Bij))[hjk(~)] = OE#~O _#~(Aij)[hjk(~)].
3 The main theorem
Let us now sketch our strategy of computing Ext *(W~(i+j), S(i)Fj). First we
k(~)
compute Ext*(D~(i+j), S(i)Fj) by manipulating with the second variable and
k(~)
using Fact 2.5. Then, by considering the resolution of the first variable we
get the general formula. Of course, the first step is more difficult because the
operation Fkjis involved here. However we have succeeded in constructing
of the "structural arrow" Fkj(_~), I was not able to conduct the proof along
the lines of the proof of [C1, Th. 4.3]. The problem was, that after lifting
8
the resolution to the level of Fkjit was difficult to show that we obtained
an exact complex in the sense of the triangulated structure in DP D~(i+j).
Thus I was forced to come back to ideas of [FLS], [FS], [FFSS], and prove
the formula inductively using the (Schur)-De-Rham complex. The difference
with [FFSS] is that we perform induction decreasing the number of twists
(the starting point is provided by [C1, Th. 5.1]).
Since the Schur-De-Rham complex will play a crucial role in the proof,
we need an analogue for complexes of some constructions and computations
we achieved in the previous section. We start by observing that we have the
"enlarged structural arrowsä lso for the Schur complexes. In order to show
j(e~)(i)
that there exists an arrow S(i)Fj-! SFk in DP (i+j)we recall that by
k(~) W~
[ABW, Cor. V.1.14] the s-th degree component of SFjk(~)has a filtration with
the graded object M
Wefi SFj(~)=fi.
|fi|=s k
Now, by Fact 2.4 and the obvious remark that a diagram has a trivial p-core
if and only if its conjugate has, we conclude that all factors in the filtration
for fi which are not of the form Fkj(ff) are trivial in the localized category.
Hence, the obstacle to the existence of the arrow (which will also be denoted
by Fkj(i)(_~)) disappears. In a similar fashion we show the existence of an
j(~)(i) (i)
arrow Fkj(i)(OE~) : Fk -! SFj . Thanks to the Exponential Formula
k(~)
we get the following "graded analogueö f some formulae of Fact 2.5
Fact 3.1 The following isomorphisms hold:
j(~)(i) ~ d j ~ 0j
Ext*(Dd(i+j), Fk )) = ~ (s (Bij))[hk(~)] = (Aij)[hk(~)],
j(~)(i) ~ d j ~ j
Ext *(Dd(i+j), SFk ) = s (s (Bij))[hk(~)] = S (Aij)[hk(~)].
Under these isomorphisms, the map induced on Ext*(Dd(i+j), -) by the com-
position Fkj(i)(_~) O Fkj(i)(OE~) is equal to _~ O OE~(Aij).
These formulae are straightforwardLconsequences of Fact 2.5 (Ext *(F, C)
stands for the sum s Ext*(F, Cs); ~ ~, s~ mean self-evident graded d-
functors).
Now we are in a position to complete the first part of our program.
9
Theorem 3.2 For any diagram ~ of weight d, the map Fkj(i)(_~) : S(i)Fj-!
j k(~)j
SFk(e~)(i)induces a monomorphism Ext*(Dd(i+j), S(i)Fj) -! Ext*(Dd(i+j), SFk(e~)*
*(i)),
k(~)
whose image is S~(Aij)[hjk(~)].
Under this identification, the map induced on Ext *(Dd(i+j)), -) by Fkj(i)(OE~)
is equal to OE~(Aij)[hjk(~)] : ~(Aij)[hjk(~)] -! S~(Aij)[hjk(~)].
Proof: We proceed by a double induction: the external on d and internal
on j. Let first j = 0. In order to determine the group Ext *(Dd(i), (S(i)~)s)
(s refers to the grading in the Schur complex) we consider a filtration on
L (i) (i)
(S(i)~)s with the graded object ffWeff S~=ff(cf. [ABW, Cor. V.1.14]). The
L u s(i) (i)
spectral sequence of this filtration has E1st= u+v=s+t Ext (D , Weff)
Extv(Dd-s(i), S(i)~=ff), where t is a position of ff in the lexicographic order*
*ing of
subdiagrams in ~ of weight s. Thanks to [C1, Th. 4.3] we know the groups
E1**. In particular, they are concentrated in even total degrees (because
Extodd(Dd(i), F (i)) = 0). Therefore, the differentials in this spectral sequen*
*ce
are trivial, and we obtain
M * (i) * (i)
Ext *(Dd(i), (S(i)~)s) = Ext (Ds(i), Weff) Ext (Dd-s(i), W~=ff)
|ff|=s
M
= (Weff S~=ff)(Ai0) = Ss~(Ai0).
|ff|=s
To get part of the theorem concerning the maps OE(i)~and _(i)~, we observe
that it follows immediately from the construction of the filtration that it is
compatible with these maps (in fact this filtration is a special case of the
filtration giving the Decomposition Formula for Schur complexes (cf. [ABW,
Cor. V.1.14])), hence we may apply the induction hypothesis. Thus we may
restrict attention to the cases s = 0 and s = d. In the first case the descrip-
tion of the maps OE(i)~*and _(i)~*easily follows from [C1, Th. 5.1]. But the ca*
*se
s = d requires some work. Namely, the fact that _#(i)~*: Ext*(Dd(i), De~(i)) -!
Ext*(Dd(i), W~(i)) and OE#(i)~*: Ext *(Dd(i), W~(i)) - ! Ext *(Dd(i), ~(i)) ar*
*e re-
spectively epic and monic follows only from ([C1], Fact 5.4). But once
we already know this, the required description follows from the fact that
(OE#(i)~O _#(i)~)* : Ext *(Dd(i), De~(i)) - ! Ext *(Dd(i), ~(i)) may be, by [C*
*1,
Th. 5.1], identified with OE#(i)~O _#(i)~(Ai0). This completes the proof for
j = 0.
10
Thus we may start our induction step. We will show our assertion for
given j, d assuming it for d0, j0 such that d0 < d, j0 j and such that
d0 d, j0 < j. Let EI, EII be respectively the first and the second spec-
tral sequence of hyperExt of Dd(i+j)with coefficients in the Schur-De-Rham
complex S(i)Fj, and let D be the spectral sequence of hyperExt of Dd(i+j)
k(~)
with coefficients in the Schur-Koszul complex S(i)Fj. We will compare these
k(~)
spectral sequences with the sequences E0I, E0II, D0 defined analogously for
j(i)(e~)
the complex SFk . Let us first examine the sequence EII. Thanks to
[C2, Th. 5.3] and the induction hypothesis, we have the group S~(Ai+1,j-1)
in the second term of this sequence, and we know that Fkj-1(i+1)(_~) in-
duces a monomorphism E2II-! E02II, which may be identified with the map
_~(Ai+1,j-1) : S~(Ai+1,j-1) - ! Se~(Ai+1,j-1). Moreover, as we remember
from the proof of [FFSS,Th. 4.5], there is only one nontrivial differential in
the sequence E0II. Hence E02IImay be viewed as a complex. This complex is
the Schur-Koszul complex Se~(ffi) associated to a sequence Ai+1,j-1:
0 -! Ai+1,j-1-ffi!Ai+1,j-1-! 0,
whose differential ffi is a differential in the sequence EII for d = 1. It eas-
ily follows from the acyclicity of a Schur-Koszul complex associated to an
identity map [ABW, Cor. V.1.5] that for a general f : V -! W we have
H*(S~(f)) = S~(f0) as graded functors where f0 : ker(f) -! coker(f) is an
arbitrary map (we recall that the vector spaces in a Schur complex depend
merely on the source and target of the map). Thus, since it was shown in
j(e~)(i)
[FS, Th. 4.5] that H*(Ai+1,j-1) = Aij, we obtain hExt *(Dd(i+j), SFk ) =
Se~(Aij), which is yet another translation of the calculations of [FFSS] into
a more invariant language. The most important consequence of this point
of view is that we still have hExt *(Dd(i+j), S(i)Fj) = S~(Aij) and we get an
k(~)j
inclusion hExt *(Dd(i+j), S(i)Fj) -! hExt *(Dd(i+j), SFk(e~)(i)).
k(~)
We now turn to the sequences EI, D. Let us look at the first terms of these
sequences. In the sth column we have the group Ext*(Dd(i+j), (S(i)Fj)s). Ap-
k(~)
plying to the second variable the filtration described in [ABW, Cor. V.1.14]
we get a spectral sequence converging to the Ext-group under considera-
tion. The first term of this sequence is E1*q= Ext *+q(Ds(i+j), W (i)g)
Fjk(ff)
11
Ext*(Dd-s(i+j), S(i)Fj ), where q is a position of ff in the lexicographic or-
k(~=ff)
dering of subdiagrams in ~ of weight s=pj (the columns with numbers nondi-
visible by pj are trivial by Fact 2.4). Let us note that the groups appearing
in this spectral sequence are known by the induction hypothesis (for Fp-1-k),
unless s 6= 0, dpj. Our task will be to show that the differentials in this se-
quence are trivial. The situation is slightly more complicated than that for
j = 0, because this time it does not suffice to observe that Aij is concen-
trated in even degrees. We should also show that the lowest Ext-degrees
of nontrivial elements in all columns have the same parity. To this end, let
us compute the smallest u + v for which there exists a nontrivial element in
Extu(Ds(i+j), W (i)g) Extv(Dd-s(i+j), W (i)j ). This degree is equal to
Fjk(ff) Fk(~=ff)
pj - 1
(s - fff- eff)(pj - 1) + eff(2(pj - 1) - (p - 1 - k)______)+
p - 1
pj - 1
+(f~ - fff)(pj - 1) + (e~ - eff)k ______.
p - 1
Now we observe that since the multiplicities of effand fffare always even,
the parity of the whole expression does not depend on ff. Therefore, we get
for all s 6= 0, dpj
M * (i) * (i)
Ext*(Dd(i+j), (S(i)Fj)s) = Ext (Ds(i+j), W ) Ext (Dd-s(i+j), W j )
k(~) |ff|=s Fgjk(ff) Fk(~=ff)
M
= (Weff S~=ff)(Aij) = Ss~(Aij).
|ff|=s
In the further part of the proof, the shifts of gradings will not play an essen*
*tial
role, so we will skip them in order to simplify notation.
j(e~)(i)
Let us put X = ker(Ext *(Dd(i+j), S(i)Fj) -! Ext*(Dd(i+j), SFk )). By
k(~)
the induction hypothesis and the previous considerations, X is concentrated
in the two extreme columns of EI and D. We will denote by X0 its part
contained in the 0th column and by X1 the part contained in the (dpj)th
column. Let @ means the differential in D. Since the sequence D converges
j
to 0, @dp (X1) X0. Thus we see that X1 must be trivial up to Ext-degree
dpj - 2. We now turn to the sequence EI. We denote by Y the subset of
the 0th column of E1Iconsisting of elements surviving to infinity. Since, as
12
we know from [FFSS, Th. 4.5], the differentials in E0Iare trivial; there are
no elements outside Y up to Ext-degree dpj - 2. It means that there are no
differentials in the sequence EI up to theLtotal degree dpj- 2. Hence, for s
dpj - 2 we have hExt s(Dd(i+j), S(i)Fj) = sq=0Extq(Dd(i+j), (S(i)j)s-q) =
L k(~) Fk(~)
s-1 s-q q s d(i+j) (i)
q=0(S~ (Aij)) Ext (D , SFjk(~)). But on the other hand, as we
remember from the analysis of the sequence EII, hExt s(Dd(i+j), S(i)Fj) =
L k(~)
s s-q q s d(i+j) (i) 0 s
q=0(S~ (Aij)) . Hence we get dim(Ext (D , SFjk(~))) = dim ((S~(Aij)) =
dim((S~(Aij))s). But since the last space is a subquotient of the first, we ob-
tain that Ext s(Dd(i+j), S(i)Fj) = (S~(Aij))s. It also means that X0 is trivial
k(~)
up to degree dpj- 2. Let us come back to the sequence D. On account of the
last sentence we get that X1 is trivial up to degree 2(dpj - 2). This, when
we turn again to EI, enables us to enlarge the range of degrees in which Y
is trivial to 2(dpj - 2). As a result we obtain the required computation of
Exts(Dd(i+j), S(i)Fj) and the triviality of X0 up to degree 2(dpj - 2). Iter-
k(~)
ating this argument (ie. strictly speaking applying the third induction, this
time on Ext-degree) we conclude that X = 0, Ext*(Dd(i+j), S(i)Fj) = S~(Aij)
k(~)
and that the differentials in EI are trivial. The last two facts also show, by
dimension counting, that Ext *(Dd(i+j), W (i)g) = We(Aij). This completes
Fjk(~) ~
the proof of the description of the groups Ext*(Dd(i+j), S(i)Fj). The last part
k(~)
of the theorem (concerning the arrow Fkj(OE~)) easily follows from the facts
we have already proved. ||
Remark: Thanks to the Exponential Formula one can immediately general-
ize Theorem 3.2 to the formula Ext *(D~(i+j), S(i)Fj) = s~(s~(Bij))[hjk(~)] =
k(~)
s~(s~(Bij))[hjk(~)] for an arbitrary diagram ~. We remind the reader that
the case of Weyl and Schur funcors is very special (see discussion after [C1,
Lemma 5.2]), and we have two alternative descriptions of the Ext-groups
here (in general, d-functors applied to left and right d-structures need
not commute (see an example given in the proof of [C1, Th. 4.3])). We will
use both descriptions: the first in the proof of Th. 3.4, the second in the
proof of Fact 3.3.
Now, we would like to generalize the computation of Ext-groups to the case
13
of an arbitrary Weyl functor in the first variable. Observe however, that the
method of the proof of Th. 3.2 does not work in this case, because it would
require the computation of Ext-groups between two Weyl functors which
does not fit our scheme. Luckily, this time we can apply the machinery de-
veloped in [C1, Sect. 3], since there are no problems with transformations of
the first variable.
Therefore, we should start with understanding the functoriality of the
computations achieved in Th. 3.2.
0
Fact 3.3 For any transformation OE : D~ - ! D~ the induced morphism
0(i+j) (i) * ~(i+j) (i) #
OE(i+j)*: Ext*(D~ , SFj ) -! Ext (D , S j ) is equal to OE (s~(Bij)).
k(~) Fk(~)
Proof: Let us consider a commutative diagram (strictly speaking coming
from the morphisms in the category DP D~(i+j) D~0(i+j))
0(i+j) (i) ffi(i+j)* * ~(i+j) (i)
Ext*(D~ , SFj ) -! Ext (D , S j )
j(i) k(~) Fk(~)
# Fk (_~)* # F j(i)(_~)*
0(i+j) Fj(e~)(i)ffi(i+j)* * ~(i+j) Fj(e~)(i)
Ext*(D~ , S k ) -! Ext (D , S k ).
Identifying known groups and arrows we get (up to shift) the diagram
0 ffi(i+j)* ~
s~ (s~(Bij)) -! s (s~(Bij))
| ~0 | ~
| s (_~(Bij)) |s (_~(Bij))
|? |?
0 ~ ffi#(s~(Bij))~~
s~ (s (Bij)) -! s (s (Bij)).
Of course, replacing of the top arrow by OE# (s~(Bij)) does not affect the
commutativity of the diagram. This, thanks to the monomorphicity of the
right vertical arrow, gives our assertion. ||
We have now all ingredients we need for the proof of our main result.
Theorem 3.4 For any diagrams ~, ~ of weight d, and any i, j, k, we have
Ext *(W~(i+j), S(i)Fj) = s~(s~(Bij))[hjk(~)] = s~(s~(Bij))[hjk(~)].
k(~)
Moreover, for any transformation OE : w~ -! w~0, the induced map OE*(i+j):
Ext*(W~(i+j)0, S(i)Fj) -! Ext*(W~(i+j), S(i)j) is equal to OE# (s~(Bij)).
k(~) Fk(~)
14
Proof: In fact, the proof consists of slightly rearranged elements of the
proofs of [C1, Th. 4.4] and [C1, Th. 5.1].
We apply the functor Ext*(-, S(i)Fj) to the (i + j)-th twisted resolution
k(~)
of W~ by divided powers starting with the structural arrow. In the resulting
complex
_#*~ * e~(i+j) ffi*1 * ~1(i+j)
0 -! Ext*(W~(i+j), SFjk(~)) -! Ext (D , SFjk(~)) -! Ext (D , SFjk(~)*
*) -!
all the groups and arrows starting with the second spot are known by The-
orem 3.2 and Fact 3.3. Thus, let us consider the commutative diagram (in
which we omit shifts)
_~(s~(Bij)) ffi#1(s~(Bij)) 1
0 -! s~(s~(Bij)) ________- s~(se~(Bij)) ________- s~(s~ (Bij)) -!
# #
_#*~ * e~(i+j) ffi*1 * ~1(i+j)
0 -! Ext*(W~(i+j), SFjk(~)) -! Ext (D , SFjk(~)) -! Ext (D , SFjk(~*
*)) -!
where the vertical arrows are isomorphisms. The proof will be finished if
we complete this diagram by the left vertical arrow. For this, it suffices to
show that the top row is exact. But this complex, if we neglect internal
grading, is isomorphic to the sequence (*) from the proof of ([C1], Th. 4.4)
for F = W~, G = S~, f = w~, g = s~.
We get the assertion concerning functoriality in a similar fashion to that
in ([C1], Th. 4.4). I leave details to the reader, for we just apply the same
d-transformations to the same d-functors as in ([C1], Sect. 4). ||
Remark: We say nothing about the functoriality with respect to the sec-
ond variable in Th. 3.4. The reason is that, of course, one cannot expect
good properties of any morphism SFjk(~)-! SFjk(~0). It seems reasonable
to consider only maps somehow "induced" by maps S~ -! S~0 , but then
we encounter a problem caused by the fact that the shape of a slice which
determines the shift in Ext-grading depends on the position of a box with
respect to the main diagonal. Let us illustrate this by a simple example. For
p = 2 we have a nontrivial transformation æ : S2 -! 2. Thus we would ex-
pect that the induced map Ext *(D2(i+1), S(i)F0((12))) -! Ext*(D2(i+1), S(i)F0(*
*(2)))
is equal to æ(Ai1). But this cannot happen because h10((12)) 6= h10((2)).
15
4 Toward the general case
We have studied in [C1] diagrams of the same weight, and we have got
the calculations of Ext-groups in terms of Ext *(I(i), I(i)). Thus we can say
that the Ext-groups "were localized in the boxes of diagrams" (we recall
how strong functoriality we obtained in [C1, Th. 5.1]). The situation con-
sidered in the previous section was more complicated. For example, the
epimorphism Ip -! Sp preserves no information about Ext *(I(i), -). Nev-
ertheless, when we deal with a diagram Fkj(~) then the information about
Ext*(W~(i+j), S(i)Fj(~)) for |~| = |~| seems to be localized in pj-slices of Fk*
*j(~)
j(i)
(we express these Ext-groups by Ext *(I(i+j), Sp )). The reason for which
the map Ip -! Sp looses information is that it tears the p-slice apart. We
had to introduce the äm ps" Fkj(OE~), Fkj(_~) in order to preserve the struc-
ture of pj-slices.
Now, we turn to the general situation. Our ultimate goal is to compute
the groups Ext *(W~(*), S(*)~) for arbitrary diagrams ~, ~. But it turns out to
be a difficult problem. The example which indicates the nature of difficul-
ties in a very striking way is just ~ = (1). It seems impossible to describe
the groups Ext*(I(a+i), S(i)~) (for a diagram ~ of weight pa) in terms of some
other already known Ext-groups, for all structural arrows loose information
since Ext *(I(i), F G) = 0 if |F |, |G| > 0. One can rather suppose that
such groups form another elementary "block" by which other Ext-groups
can be expressed. According to this point of view, Ext *(I(i), I(i)) is just
the simplest example of these blocks. But the groups Ext *(I(i), I(i)) were
computed in [FS, Th. 4.5], what made all further calculations possible. So,
let us check whether the method used in [FS] can be applied to computing
Ext*(I(a+i), S(i)~) in general. The proof of [FS, Th. 4.5] consisted of two ste*
*ps.
i
The first was computation of Ext *(I(i), Sp ), which was easy, because of the
i
injectivity of Sp . The second (and main) was induction on j which allowed
i-j(j)
to compute Ext *(I(i), Sp ) for larger and larger j (till j = i). Here, the
crucial role was played by analysis of hyperExt-groups with coefficients in
the Koszul and De-Rham complexes. It turns out that the second part of
this procedure carries over to the general situation.
Fact 4.1 Let ~ be a diagram of weight pa and let G = Ext *(I(a+i), SFik(~)).
16
Then a+i
Ext*(I(a+i), S(i)~) = G k[x]=(xp )[-hik(~)],
where |x| = 2pa.
Proof: Let Gj = Ext *(I(a+i), S(i-j)Fj). It suffices if we show that Gj-1 =
k(~)
Gj k[y]=(yp)[-h1k(Fkj(~)], where |y| = 2pa+j-1.
Let K (i-j)jdenote the quotient of S(i-j)Fjconsisting of boundaries of the
k(~)
differential in the Schur-Koszul complex (the reader of [FLS] and [FS] rec-
ognizes the familiar strategy). By [C2,Th. 5.3] and the arguments of [FLS,
proof of Prop. 3.5], H*(K (i-j)j) = K (i-j+1)j-1. We denote by EI, EII respec-
tively the first and second spectral sequence converging to hExt *(I(a+i), K j).
Since the Schur-Koszul complex is exact, we have a short exact sequence of
complexes
0 -! K (i-j)j[-1] -! S(i-j)Fj-! K (i-j)j-! 0.
k(~)
Thus we see that in the sth column in the first term of EI we have Gj[s - 1],
while in the sth column in the second term of EII for s h1k(Fkj(~)) we
have Gj-1[s - 1 - h1k(Fkj(~))]. Dimension counting shows that the proof will
be finished if we show that the differentials in these spectral sequences are
trivial. Indeed, if we would show this, then, since both sequences have a
common limit, we would get
a+j pa+j-1 1 j
Gj k[z]=(zp ) = Gj-1 k[z]=(z )[hk(Fk(~))],
where |z| = 2, which gives our assertion. Thus it remains to show the
triviality of the differentials.
We first show it for EII. Let Aq : E2II,*q-! E2II,*+1,q+1be the con-
necting homomorphism in the long exact sequence of Ext-groups induced
by K (i-j+1)j-1[-1] -! S(i-j+1)Fj-1-! K (i-j+1)j-1(this map gives an isomorphism
k (~)
(shifting degree) between columns of E2II). Since the maps inducing A* sum
up to the morphism of complexes, the maps A* commute (up to sign) with
differentials in EII (when we look at the hyperExt spectral sequences as
sequences of suitable filtrations, we may derive this fact from the classical
observation in [CE, Prop. IV.2.1]). But since these differentials are of type
(r, -r +1), we immediately conclude that they must be trivial, because there
is only a finite number of nonzero columns in EII.
17
For the sequence EI this argument is unsufficient because the differentials
are of type (-r + 1, r). Using it we can only conclude that if there is some
nontrivial differential then there must be a nontrivial differential arriving to
the last column. But to show the triviality of differentials coming to the
last column we will use a different argument. We take some x 2 E1I,n,pa+j-1
and assume that 0 6= y = @1(x) (@ stands for the differential in EI), where
y 2 E1I,n,pa+j= Extn(I(a+i), W (i-j)g).
Fjk(~)
Let us consider a twisted "resolutionö f S(i-j)Fjby symmetric powers
k(~)
0(i-j)
0 -! S(i-j)Fj-! S~ - ! . . .
k(~)
We get it by extension of a resolution of S(i-j)Fjto whole Schur complexes,
k(~)
which is possible because each transformation between products of symmet-
ric powers is a composition of transformations of three simple types (cf. [C1,
proof of Lemma 3.4]), and these simplest transformations are obviously ex-
tendable. I put the word resolution into quotation marks, because I do not
claim that the resulting sequence of complexes is exact (which is probably
true, but we need not this fact). The important thing is that we do have
an exact sequence in the highest degree component of this "resolution". In-
*(i-j)pa+j
deed, as we remember from [C1, Cor. 5.5], the sequence (S~ ) of the
highest degree components in the "resolution" is the öK szul dualö f the
0(i-j) (i-j)
sequence 0 - ! S(i-j)Fj-! S~ -!. Thus it is a resolution of W
k(~) Fgjk(~)
by twisted exterior powers. Now, when we consider the spectral sequence
converging to hyperExt of I(a+i)with coefficients in the complex of bound-
*(i-j)pa+j
aries of the Schur-Koszul complex (S~ ) (this sequence remains exact
by the arguments of [FLS, proof of Prop. 3.5]), then, of course, there exists
s(i-j) s
0 6= z 2 Ext *(I(a+i), S~ ) such that z = ffi (y), where ffi denotes the di*
*f-
ferential in this spectral sequence. Let @0 denote the differential in the first
spectral sequence converging to hExt *(I(a+i), -) with coefficients in the com-
s
plex of boundaries of the differential in the Schur-Koszul complex S~ . Since
the maps in the"resolutionö f S(i-j)Fjare morphisms of complexes, we have
k(~)
ffi O @ = @0 O ffi (we use again [CE, Prop. IV.2.1]). But according to the
calculations of [FS, Th. 4.5], ffi0 = 0 by dimension argument. Hence we get
z = 0 which leads to a contradiction finishing the proof of the triviality of
differential @1 in EI. We prove the triviality of higher differentials in the
18
same manner. ||
Remark: Comparing this proof with that of [FS, Th. 4.5], we find a
new ingredient: use of maps A*. This trick would also allow to simplify the
proof of that theorem.
Thus our task is reduced to computing Ext *(I(a+i), SFik(~)). Unfortunately,
in general SFik(~)is not injective and, surprisingly enough, the computation
of these groups becomes the main problem. In the remainder of this section
I will sketch some approach to this problem, which at least suggests the
language in which the answer should be given. But first of all we shall show
that the groups under consideration may be nontrivial for ~ not being a pa-
hook (for pa hooks the computation is easy). This vanishing would be quite
reasonable in light of the correspondence between the twisting of the first
variable and enlarging a diagram in the second (eg. this is true for a = 1
where Ext*(I(1), S~) = 0 for ~ not being a p-hook). But the simplest possible
example shows that this is not the case
Fact 4.2 For p = 2,
(
1 n = 1, 2
dim (Ext n(I(2), S(2,2))) =
0 otherwise.
Proof: We consider the spectral sequence converging to Ext *(I(2), S(2,1)
S(1)) = 0 provided by the Littlewood-Richardson decomposition of the sec-
ond variable [Bo]. In the columns of E1 we have Ext*(I(2), S(3,1)) = Ext2(I(2),*
* S(3,1)) =
k, Ext *(I(2), S(2,2)) and Ext *(I(2), S(2,12)) = Ext 1(I(2), S(2,12)) = k. Th*
*us we
see, that there are two possibilities: either the groups we are interested in
have the description predicted by our assertion, or they are trivial. To rule
out the second possibility, we consider the spectral sequences converging to
hExt*(I(2), S(2,2)) (we take the Schur-de-Rham complex here). If our groups
were trivial, then the whole first term of the first spectral sequence would
be trivial (the triviality of all columns except the first and last one follows
from Ext *(I(2)), F G) = 0 for |F |, |G| > 0; hence the triviality of the last
column follows from the exactness of the Schur-Koszul complex). Therefore
the second spectral sequence would converge to 0. But according to [C2,
Fact 8.1], the second term of this sequence has four nontrivial columns in
which we have the groups Ext*(I(2), 2(1)). This gives a contradiction which
19
finishes the proof. ||
In fact, it is possible to compute groups Ext *(I(a), S~) also for some other
small diagrams ~ using such ad hoc methods. But I think it will be more
interesting to sketch the general approach to the problem.
j(a-j)
The main idea is that since one can hope that Ext *(Dp , S~) =
j(a-j) * pj(a-j)
Ext*(Ip , S~) pj, one can try to compute Ext (D , S~) by induction
on j using the Koszul and De-Rham complexes. But to start this program
j(a-j)
we should first compute the groups Ext *(Ip , S~). Already this prob-
lem turns out to be highly nontrivial. Let us focus on the case j = a - 1.
It will be convenient to consider a slightly more general problem, namely
that of computing Ext *(Id(1), S~) for a diagram ~ of weight dp. By the ear-
lier considerations this group is trivial if ~ has a nontrivial p-core; and if
~ = Fk(~0) then, according to Th. 3.4, Ext *(Id(1), S~) = s~0(B01)[h1k(~0)] =
s~0(k[ d])[h1k(~0)] = Sp~0[h1k(~0)] (we recall that Sp~ means the Specht mod-
ule associated to the diagram ~ (cf. [C1, Sect. 3]). In order to understand the
situation of a diagram with a trivial core but the quotient consisting of sev-
eral diagrams we shall develop notation introduced in [C2, Sect. 4]. We say
that R = {~ = ff0 ff1 . . .ffd = ;} is a decomposition of ~ into slices
if for every 1 s d, the diagram ffs is obtained from ffs-1 by removing a
rim p-hook. In such a situation we call skew hooks Øs = ffs-1 \ ffs, the slices
of this decomposition. We recall from [C2, Sect. 4] that it may happen that
different decompositions produce the same set of slices (in fact this is always
the case if ~ = Fk(~0)). Let us come back to Ext-groups. We consider the
Decomposition Spectral Sequence [C1, Sect. 2] converging to Ext*(Id(1), S~).
Columns of the first term of this sequence are labeled by decompositions of
~ into slices. In the column corresponding to a decomposition R with the set
of slices {Ø1, . .,.Ød} we have
Ext *(I(1), Sffl1) . . .Ext*(I(1), Sffld) = k[h(R)],
P i i
where the shift h(R) is given by the formula h(R) = ih(Ø ), and h(Ø )
is equal to the number of columns in Øi minus 1. We shall show that the
differentials in this spectral sequence are trivial. To do this, it suffices to
show that all the numbers h(R) have the same parity. But the last statement
follows form the fact that the number (-1)h(R) is equal to "the sign of the
permutation taking the natural ordering of beads before moving them to
the configuration corresponding to the core and after it" [JK, p. 81]. For the
20
reader who does not like beads and runners we can can give a less elementary
argument. Namely, it follows from the Muranghan-Nakayama formula [JK,
p. 60] that the value of the characterPof Sp~ on a permutation being a sum
of d cycles of length p equals R(-1)h(R). On the other hand this value is
computed on p. 83 as f(~) where f(~) is the number of decompositions of
~ into slices. Therefore all the numbers (-1)h(R) must be equal (of course,
beads are hidden in the proof of the formula on p. 83). Thus we have shown
that
dim(Ext *(Id(1), S~)) = f(~).
Moreover, it immediately follows from [JK, Th. 7.27] that for ~ with a trivial
p-core and the p-quotient (q0(~), . .,.qp-1(~)) we have
f(~) = dim ((Spq0(~) . . .Spqp-1(~)) " d).
This formula is attractive, because it suggests the structure of a d-module
on Ext *(Id(1), S~), which we should understand, since the next step in our
program will be taking the coinvariants. Unfortunately, it is unlikely that
there is an isomorphism of d-modules Ext *(Id(1), S~) ' (Spq0(~) . . .
Spqp-1(~)) " d, because the left-hand side is a sum of its homogeneous
components, while (it seems that) there is no such a decomposition of the
right-hand side. But some numerical experiments suggest that, at least in
some special cases, the situation is even simpler.
Conjecture 4.3 There is an isomorphism of d-modules
M
Ext*(Id(1), S~) ' Nff;q0(~),...,qp-1(~)Spff,
ff
where Nff;q0(~),...,qp-1(~)is the multiplicity of Spffin the Littlewood-Richard*
*son
decomposition of (Spq0(~) . . .Spqp-1(~)) " d.
The proof of this conjecture would require a thorough understanding of an in-
terplay between the Decomposition Formula and the Littlewood-Richardson
rule.
Acknowledgements. I wish to thank Stanis_law Betley for many stimulating
discussions on functor category and careful reading the preliminary version
of this article.
21
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22