Table Of ContentGaussian maps and plethysm
Laurent Manivel
October 1992
1. Introduction and main results.
Plethysm was introduced by Littlewood [11] in the late thirties: it is nothing else
than the composition of representations of the complex linear group (or, equivalently,
of symmetric functions). The problem is to get some understanding of the irreducible
components of such a composition, and of their multiplicities.
The maingoal of this paper is to shed somelight onto one of the remarkableproperties
of plethysm: the stability of its multiplicities. To explain what we mean by this, consider
k l
for some complex vector space V the composition of symmetric powers S (S V), which
we shall mainlybe interested in. Let (cid:23) = ((cid:23)1;:::;(cid:23)m) be an increasing partition of kl: we
denote its weight by j(cid:23)j, which is the sum of its parts, its length by l((cid:23)), the number of
its nonzero parts, and its height by h((cid:23)), that is, its largest part. For reasons that should
become clear very soon, we will denote the multiplicity of the irreducible Gl(V)-module
(cid:23) k l k;l
S V of dominantweight (cid:23), in the plethysmS (S V), by (cid:23) ((cid:22)), where (cid:22) = ((cid:23)1;:::;(cid:23)m(cid:0)1)
is the partition one obtains by deleting the largest part of (cid:23) (vice versa, we will denote
k;l
(cid:23) = ((cid:22);kl(cid:0)j(cid:22)j)). This multiplicity(cid:23) ((cid:22)), as a function of k and l, is known to stabilize
when l becomes large enough; more precisely as soon as l (cid:21) h((cid:22)) ([14], [5]).
In this paper, we shall prove that stability also occurs with respect to k, and give a
formulafor k-stable multiplicities (seetheorem2.3.1, corollary 2.3.2, proposition 3.1.1 and
theorem 4.1.1):
k;l (cid:22);kl(cid:0)j(cid:22)j k l
Theorem The multiplicity (cid:23) ((cid:22)) of S V inside S (S V) is an increasing and
stationary function of both integers k and l, and the k-stable multiplicity
1;l k;l
(cid:23) ((cid:22)) = lim (cid:23) ((cid:22))
k!1
(cid:22)
equals the multiplicity of S V in the sum
M
(cid:25)2 2 (cid:25)l l
S (S V)(cid:10)(cid:1)(cid:1)(cid:1)(cid:10)S (S V):
(cid:25)=(2(cid:25)2(cid:1)(cid:1)(cid:1)l(cid:25)l)
This stable multiplicity is achieved as soon as k (cid:21) j(cid:22)j.
1
Gaussian maps and plethysm 2
We willgive expressions of the l-stable multiplicitiesas well (theorem4.2.2), and show
that these expressions have some remarkable consequences. Recall that, following the
l k k l
celebrated Hermite'slaw of reciprocity, S (S V) and S (S V) have the same components
of length at most two. From the preceeding theorem follows an asymptotic analogue of
this reciprocity, for components of arbitrary length (theorem 4.3.1):
Law of asymptotic reciprocity Let (cid:22) be some partition; if k and l are at least equal
(cid:22);kl(cid:0)j(cid:22)j l k k l
to its weight, S V has the same multiplicity inside S (S V) and S (S V).
Moreover, stable multiplicitiescontain non trivial informations about the vanishing of
the multiplicities of a plethysm, for which we will give a simple criterion (corollary 4.4.1)
that widely generalizes the observation, (cid:12)rst made by Chen, Garsia and Remmel[6], that
no hook, other than the vertical one corresponding to the symmetric power of exponent
k l
kl, can have nonzero multiplicity in the composition S (S V).
We shall also give a partial positive answer to a conjecture of Weintraub, following
2(cid:23) k 2l
which S V should have positive multiplicity inside S (S V) as soon as j(cid:23)j = kl and
l((cid:23)) (cid:20) k (proposition 3.1.2).
Proposition Let (cid:22) be a partition of length l((cid:22)) < k, and suppose that l (cid:21) h((cid:22)). If
2(cid:23) k 2l
(cid:23) = ((cid:22);kl(cid:0)j(cid:22)j), then S V has positive multiplicity inside S (S V).
Recall that following Weintraub, multiplicities stabilize for l (cid:21) h((cid:22)): so that we have
proved Weintraub's conjecture "after stabilization" .
k l
In a (cid:12)nal section, we will obtain similar formulas for stable multiplicities of S (^ V)
(theorem 5.3.1), and derive reciprocity properties of a somewhat di(cid:11)erent nature than
the preceeding asymptotic reciprocity law (see corollary 5.3.2 and proposition 4.3.2 for a
slightly more general statement):
Theorem A. Let (cid:22) and (cid:23) be two partitions such that l (cid:21) l((cid:22));l((cid:23)), and k (cid:21) j(cid:22)j;j(cid:23)j.
(cid:22);(k)l(cid:0)(cid:23)^ (cid:23);(k)l(cid:0)(cid:22)^ k l
Then S V and S V have the same multiplicity in S (^ V).
(cid:22);2kl(cid:0)j(cid:22)j
B.Let(cid:22) be a partitionsuch that 2l (cid:21) h((cid:22)). Then the irreducible components S V
(2l)k(cid:0)1(cid:0)(cid:22)^;2l+j(cid:22)j k 2l
and S V have the same multiplicity in S (S V).
l
Here, we denoted by (k) (cid:0) (cid:23)^ the partition (k (cid:0) (cid:23)l((cid:23));:::;k (cid:0) (cid:23)1;k;:::;k) of length
k 2l+1
l. Note that the second part of this theorem is certainly untrue for S (S V). The
Young diagrams of the partitions the multiplicities of which we proved to be equal are
the following:
2kl(cid:0)j(cid:22)j 2l j(cid:22)j
z }| { z }| {z}|{
(cid:26)
(cid:22)
Gaussian maps and plethysm 3
Wethusrelatedthemultiplicitiesofpartitionsclosetobevertical,tothoseofpartitions
close to be rectangular of sides 2l(cid:2)k.
Our approach of the composition of symmetric powers is of geometric (cid:13)avour, and
followsJonathan Wahl'sidea[13]touseGaussian mapsonhomogeneousspacestorecover,
in an elementary way, di(cid:11)erent results of the representation theory of complex reductive
groups. This geometric approach is made possible mainly by Borel-Weil's theory, which
realizes irreducible representations of the linear group, as far as we are concerned in
that paper, as spaces of global holomorphic sections of homogeneous line bundles on (cid:13)ag
varieties [1].
(cid:3)
Let P(V ) be the projective space of hyperplanes of the complex vector space V: its
Picard group has for generator the quotient line bundle OV(1), the l-th power of which,
l
OV(l), has the symmetric power S V as Gl(V)-module of global sections.
l
The k-th tensor power of S V thus identi(cid:12)es with the space of global sections of the
(cid:3) (k)
total exterior product of k copies of OV(l), on the product variety P(V ) of k copies
(cid:3)
of the projective space P(V ). But that product variety contains a diagonal (cid:1), which
k l l (cid:10)k
provides us with a (cid:12)ltration of the space of sections S (S V) (cid:26) (S V) , by the vanishing
order on that subvariety. Will shall prove that the i-th quotient of that diagonal (cid:12)ltration
is the sum M
k;l k;l (cid:22);kl(cid:0)j(cid:22)j
G = (cid:23) ((cid:22))S V:
i
j(cid:22)j=i
k;l
This geometric interpretation of the multiplicities (cid:23) ((cid:22)) in mind, we will be able to give
extremelysimpleproofs of some of its properties which do not appear completelyobvious
in the pure representation theoretic context. Moreover, when k (cid:21) i, we will construct
enough symmetricsections vanishing at order i on (cid:1) to get a complete description of the
i (cid:3) (cid:3)
homogeneous subbundle of S N (cid:10)L generated by the i-th jets of such sections, where N
is the conormal bundle of the diagonal. Our formulas for stable multiplicities will follow
from that analysis.
I would like to mention that these formulas were generalized by Michel Brion [2], to
thecase of thecomposition of arbitrary representations of the lineargroup, for which they
unfortunately become rather inextricable. J.Y. Thibon also informed us he was able to
recover them from vertex operators and the ideas developped in [5].
2. The diagonal filtration.
2.1 Representations of the linear group.
We will make use in this paper of two di(cid:11)erent realizations of an irreducible represen-
tation of Gl(V), the linear group of automorphismsof a (cid:12)nite dimensionalcomplexvector
space V.
Gaussian maps and plethysm 4
The (cid:12)rst one is given by the Weyl construction [15]: the symmetric group Sk acts on
(cid:10)k (cid:21)
V , and the irreducible Gl(V)-module S V is the image of some Young symmetrizer
de(cid:12)ned by the partition (cid:21) of k.
(cid:21)
The second one is more geometric and identi(cid:12)es, following Borel, Weil and Bott, S V
with the module of global regular sections, on some (cid:13)ag variety of V, of the line bundle
de(cid:12)ned by an adequate character of the corresponding parabolic subgroup of Gl(V) [1].
Themostfamiliarexampleof this constructionis theusual identi(cid:12)cationof thesymmetric
l (cid:3)
power S V with the Gl(V)-module of global sections of the line bundle OV(l) over P(V ).
(cid:21) l
Combining both approaches, we will give to the plethysm S (S V) the following ge-
l (cid:10)k
ometric content: (S V) will be identi(cid:12)ed with the module of global sections, on the
(cid:3)
product X of k copies of P(V ), of the total exterior product L of k copies of OV(l); and
(cid:21) l
S (S V) with the image of that tensor power by some Young symmetrizerde(cid:12)ned by the
partition (cid:21). We shall on occasion call its elements (cid:21)-symmetric sections.
2.2 Gaussian maps.
(cid:3)
Let(cid:1) be thediagonal inX. The restrictionof L to (cid:1),naturally identi(cid:12)edwithP(V ),
is the line bundle OV(kl). Moreover, the restriction map
(cid:21) l l (cid:10)k kl
S (S V) (cid:26) (S V) = (cid:0)(X;L) (cid:0)! (cid:0)((cid:1);Lj(cid:1)) = S V
is zero, because of the skew-symmetries generated by the Young symmetrizer, if (cid:21) is
not the partition of k of length one. In the opposite case, this restriction map is the
k l kl
epimorphism from S (S V) onto S V given by the Cartan product.
(cid:21);l (cid:21) l
More generally, let us denote by (cid:1) the sub-Gl(V)-module of sections of S (S V)
i
vanishing on (cid:1) at order at least i. If N denotes the normal bundle of the diagonal in X,
associating to such a section its i-th jet in directions normal to (cid:1) allows to de(cid:12)ne the i-th
Gaussian map [13]:
(cid:21);l i (cid:3)
(cid:1) (cid:0)! (cid:0)((cid:1);S N (cid:10)L);
i
(cid:21);l
the kernel of which is precisely (cid:1) . Note that the normal exact sequence
i+1
(cid:8)k
0 (cid:0)! T(cid:1) (cid:0)! TXj(cid:1) ' T(cid:1) (cid:0)! N (cid:0)! 0
is split.
(cid:21);l i (cid:3)
Let G be the homogeneous subbundle of S N (cid:10) L generated by jets of sections
i
(cid:21);l
vanishing at order at least i on the diagonal. There are natural integers (cid:23) ((cid:22)) such that
M
(cid:21);l (cid:21);l (cid:22) 1
Gi ' (cid:23) ((cid:22))S (cid:10)P(V(cid:3)) (cid:10)OV(kl):
j(cid:22)j=i
(cid:21) l (cid:21);l
We shall call the diagonal (cid:12)ltration of S (S V) its (cid:12)ltration de(cid:12)ned by the (cid:1) , and
i
denote by
(cid:21);l (cid:21);l (cid:21);l (cid:21);l
G = (cid:0)((cid:1);G ) = (cid:1) =(cid:1)
i i i i+1
the i-th quotient of that (cid:12)ltration, which is the image of the i-th Gaussian map. The
linear group being reductive, this quotient is a direct factor as well. Moreover, it can be
computed by the following straightforward consequence of Bott's theorem :
Gaussian maps and plethysm 5
Theorem 2.2.1 For every partition (cid:22), and every integer m, there is an isomorphism of
Gl(V)-modules
(cid:3) (cid:22) 1 (cid:22);m(cid:0)j(cid:22)j
(cid:0)(P(V );S (cid:10)P(V(cid:3)) (cid:10)OV(m)) ' S V:
(cid:22);m(cid:0)j(cid:22)j
Recall that our partitions increase from left to right: S V is the irreducible
Gl(V)-module associated with the partition of m, if it is one, obtained by adding to (cid:22)
the integer m(cid:0)j(cid:22)j, on its right; if this is not a partition, that is, if m < j(cid:22)j+h((cid:22)), we
simply get the zero module.
We thus get the decomposition
M
(cid:21) l (cid:21);l (cid:22);kl(cid:0)j(cid:22)j
S (S V) = (cid:23) ((cid:22))S V:
(cid:22)
(cid:21);l
In this sum, the multiplicities (cid:23) ((cid:22)) must be considered as independent of V, the
problemof decomposing a plethysmbeing "universal" with respect to V. Otherwise said,
we always assume V to be of arbitrary large dimension.
Note that we could have de(cid:12)ned the diagonal (cid:12)ltration a priori, by the height of parti-
(cid:21) l
tions involvedin the decomposition of the plethysmS (S V) into irreduciblecomponents.
Having at our disposal a simple geometric interpretation of this (cid:12)ltration, we will be able
to derive several non trivial of its properties. As a (cid:12)rst illustration, we shall now give a
very simple proof of a conjecture of Foulkes.
2.3 Jets of sections vanishing on the diagonal
k
Our approach in the sequel will be the following. We shall choose a point m on the
(cid:3)
diagonal (cid:1), corresponding to some given point m in P(V ), and some linear coordinates
(z0;:::;zn) on V such that m = (1;0;:::;0). The rational functions xi = zi=z0 then
k
provide us with coordinates in a neighbourhood of m, hence in a neighbourhood of m
(j) (j) (j)
in X. We denote these coordinates by xi = zi =z0 : the superior index numbers the
(cid:3)
di(cid:11)erent copies of P(V ) of which X is the product.
(cid:21) l (j)
An elementof S (S V) identi(cid:12)es with some homogeneous polynomial of the z , with
i
(j) (j) k
degree l in each group of variables z0 ;:::;zn . Around m , such an element is the
(j)
product of a polynomial P in the x , with the local generator of L
i
k
Y
(j) l
(cid:16) = ( z ) :
0
j=1
If this element vanishes at order i on (cid:1), we can compute the i-th term of its Taylor
k
expansion at m by di(cid:11)erentiating i times the polynomial P. Its normal i-th jet to (cid:1) is
Pn (j)
then the equivalence class of this di(cid:11)erential, modulo multiples of the sums j=1dxi ,
1
1 (cid:20) i (cid:20) n, which span (cid:10) .
(cid:1)
k i (cid:3)
At the point m , these jets will generate some linear subspace of the (cid:12)ber of S N (cid:10)L
which, since it does not depend of the system of coordinates chosen, must be invariant
Gaussian maps and plethysm 6
1
undertheaction of Gl((cid:10) ). SinceGl(V) actshomogeneouslyon (cid:1), thissubspace mustbe
(cid:1)
(cid:21);l k (cid:21);l
the(cid:12)berofG atm . Inparticular,themultiplicities(cid:23) ((cid:22))canbecomputedponctually.
i
(cid:21) l+1
Note that the polynomial P we introduced also de(cid:12)nes an elementof S (S V), that
(cid:21) l
also vanishes at order i on (cid:1), and has the same jet that the element of S (S V) which
de(cid:12)ned P, except that the local generator of L has changed. Indeed, the restriction of L
to (cid:1) is no longer OV(kl), but OV(kl+k): that amounts to a change of the largest parts
of the partitions involved in the plethysm we deal with. Simply stated, this is equivalent
to the following old conjecture of Foulkes [7]:
(cid:21);l
Theorem 2.3.1 The multiplicities (cid:23) ((cid:22)) increase with l.
As a consequence, we get an elementary proof of the stabilization of multiplicities
(cid:21);l
(cid:23) ((cid:22)), which was proved by Weintraub [14] and Carr(cid:19)e-Thibon [5]:
(cid:21);l
Corollary 2.3.2 The multiplicities (cid:23) ((cid:22)) stabilize for large values of l.
(cid:21);l (cid:22) 1 i (cid:3)
Proof: This multiplicity (cid:23) ((cid:22)) is at most that of S (cid:10)P(V(cid:3)) in S N , which does not
depend on l. 2
2.4 The (cid:12)rst non zero quotient of the (cid:12)ltration.
k l
We recalled in paragraph 2.2 that the image of S (S V) by restriction to the diagonal
kl (cid:21) l
was S V; but this restriction map vanishes for every other plethysmS (S V). The prob-
lemwewillbeconcernedwithhereisthefollowing: what istheminimalorderof vanishing
on the diagonal, of (cid:21)-symmetric sections? Otherwise said, what is the maximal height of
(cid:21) l
partitions occuring in the decomposition of S (S V) into irreducible components?
(cid:21) l (cid:25)
Theorem 2.4.1 The plethysm S (S V) has no component S V such that h((cid:25)) > h((cid:21))+
(l(cid:0)1)j(cid:21)j. Moreover, the only partition of height h((cid:21))+(l(cid:0)1)j(cid:21)j of its decomposition is
(cid:21)+((l(cid:0)1)j(cid:21)j), with multiplicity one.
Here we denoted by (m) the vertical partition of m of length one.
k l
Example: The plethysm ^ (S V) has exactly one component of height kl(cid:0)k+1, which
is a hook of length k, and no component of greater height.
Before giving a proof of this theorem, let us recall some details on the Weyl construc-
tion we alluded to in 2.1. A partition (cid:21) of an integer k can be represented by its Young or
Ferrers diagram. To every numbering of this diagram by the integers from one to k, does
correspond a Young symmetrizer,which is an element of the algebra CSk of the group of
permutations of k objects. We will choose the following numbering of the Young diagram
of (cid:21): beginning with one in its upper left corner, we will proceed from top to bottom and
from left to right, in this order (which is not the usual one).
Gaussian maps and plethysm 7
Let R(cid:21) (respectively C(cid:21)) be the set of permutations in Sk which leaves every row
(respectively every column) of the diagram of (cid:21), globally unchanged. Then c(cid:21) = b(cid:21)a(cid:21) is
a Young symmetrizer,where
X X
b(cid:21) = (cid:15)((cid:27))(cid:27); a(cid:21) = (cid:28):
(cid:27)2C(cid:21) (cid:28)2R(cid:21)
Moreover, the image of c(cid:21), acting on the k-th tensor product of a complex vector
space, is an irreducible representation of its linear group, with dominant weight (cid:21).
Proof: Let s be a section of L, which we write as the product of the local generator (cid:16) by
P J (J)
some homogeneous polynomial Q;J sQxQ of degree i, where Q and J are multiindices
(J) (j1) (ji) (cid:21) l
of the same length i, and xQ = xq1 (cid:1)(cid:1)(cid:1)xqi . The plethysm S (S V) is generated by the
images of such sections of L under the action of the Young symmetrizer c(cid:21). We have
X X X
((cid:28)(J)) ((cid:27)(J))
c(cid:21)s = sQ (cid:15)((cid:27))xQ (cid:2)(cid:16);
Q;J (cid:28)2R(cid:21) (cid:27)2C(cid:21)
and the sum over C(cid:21) factors as a product of sums, one for each column of the Young
(L)
diagram of (cid:21). But suppose some monomial x has to be completely skewsymmetrized
P
in m variables: the result will be zero if L is of length smaller than m(cid:0)1. So that c(cid:21)s
will certainly vanish if s has degree smaller than
h((cid:21))
X
(cid:3)
i(cid:21) = ((cid:21)j (cid:0)1) = j(cid:21)j(cid:0)h((cid:21));
j=1
(cid:3)
where (cid:21) is the conjugate partition of (cid:21). Moreover, if the degree i of s is at least equal
0 0
to i(cid:21), and if s vanishes at order i < i on the diagonal, its i-th jet will certainly vanish
k
at point m , which is the center of our coordinate system. The (cid:12)rst part of our claim
follows.
(cid:3)
Now suppose that s has degree i(cid:21): for c(cid:21)s to be nonzero, s must have degree (cid:21)j (cid:0)1
in the indices of the j-th column of the diagram of (cid:21), and skewsymmetrizingby elements
(cid:3)
of C(cid:21) will give something nonzero only when J has exactly (cid:21)j (cid:0) 1 pairwise di(cid:11)erent
components in this j-th column: such a multiindex is determined by the choice of one
element in each column. The corresponding skewsymmetric polynomial can be written,
for example for the (cid:12)rst column, as
X(cid:21)(cid:3)1 X
( ((cid:0)1)(cid:21)(cid:3)1(cid:0)j(cid:28)(cid:21)(cid:3)1;j)( (cid:15)((cid:26))dx(q(cid:26)1(1))(cid:1)(cid:1)(cid:1)dxq((cid:26)(cid:21)((cid:3)1(cid:21)(cid:0)(cid:3)11(cid:0)1)));
j=1 (cid:26)2S(cid:21)(cid:3)1(cid:0)1
where (cid:28)p;q is the transposition of indices p and q. Observe that the second parenthesis in
that expression corresponds to a skewsymmetrization by the (cid:12)rst column deprived of its
largest element.
Let (cid:22) be the partition one obtains after deleting the largest element of each column
of the Young diagram of (cid:21). The point now is that when we symmetrize the coe(cid:14)cients
Gaussian maps and plethysm 8
of s relatively to the di(cid:11)erent rows of the diagram of (cid:22), we get for its i(cid:21)-th di(cid:11)erential an
expression which is automatically symmetricrelativelyto the rows of (cid:21). This means that
if we associate to c(cid:21)s the product by (cid:16) of the polynomial obtained from its di(cid:11)erential
of order i(cid:21), by keeping only the monomials indexed by integers of the numbering of
the diagram of (cid:22), we get an isomorphism between the span of the i(cid:21)-jets of (cid:21)-symmetric
sections,andtheimageoftheYoungsymmetrizerde(cid:12)nedby(cid:22)and itsinducednumbering.
Otherwise said, the di(cid:11)erentials of order i(cid:21) of (cid:21)-symmetric sections vanishing at that
order on the diagonal generate at point mk a subspace of Si(cid:21)(cid:10)1 (cid:10)L which is isomorphic
X
(cid:22) 1
to S (cid:10)P(V(cid:3)) (cid:10)L. Since it is irreducible, and since the (cid:21)-symmetric sections of degree i(cid:21)
cannot vanish at order i(cid:21) + 1 on (cid:1) without being identically zero, the same conclusion
must hold for their jets. The second part of our claim then follows from Bott's theorem
and the simple remark that (cid:22) is the partition obtained by deleting the (cid:12)rst row of (cid:21). 2
3. The composition of symmetric powers.
We will pursue our study much further in the case of a composition of symmetric
k l
powers S (S V), that is, of sections of the line bundle L on X, invariant under the action
of the full symmetric group Sk. This invariance will be re(cid:13)ected by certain symmetries
readable on the jets of such symmetric sections vanishing on the diagonal. For k large
enough, we will construct enough of these to describe explicitely the homogenous bundle
they generate.
3.1 On a conjecture of Weintraub.
k l
Letusbeginwithasimpleobservation: supposewehaveasectionofS (S V)vanishing
at order i on the diagonal. Such a section is a symmetric polynomial in k groups of
k+1 l
variables; if we symmetrizeit in k+1 groups of variables, we get a section of S (S V),
still vanishing at order i on the diagonal, with an i-th jet we could derive from the same
symmetrization of the i-th jet of the original section. Hence the following analogue of
theorem 2.3.1:
k;l
Proposition 3.1.1 The multiplicities (cid:23) ((cid:22)) increase with k.
It is a conjecture of Weintraub [14] that if (cid:23) is a partition of kl of length at most k,
2(cid:23) k 2l
S V has positive multiplicityinside S (S V). We shall prove that conjecture under the
hypothesis that the second largest part of (cid:23) does not exceed l. Note that it is exactly
the hypothesis under which Weintraub proves that multiplicities stabilize: so that the
followingproposition establishes Weintraub'sconjecture "after stabilization" with respect
to l.
Proposition 3.1.2 Let (cid:22) be a partition such that l((cid:22)) < k and h((cid:22)) (cid:20) l, and let
2(cid:23) k 2l
(cid:23) = ((cid:22);kl(cid:0)j(cid:22)j). Then S V has positive multiplicity in S (S V).
Gaussian maps and plethysm 9
Proof: We proceed by induction on h((cid:22)), our claim being trivial for (cid:22) = 0. It follows
from a theorem of Newell [12] that for any partition (cid:26), the equality
p;q p;q+2 p(cid:0)1
(cid:23) ((cid:26)) = (cid:23) ((cid:26)+(2) )
p(cid:0)1
holds, where (2) is the partition which has exactly p(cid:0)1 positive parts, all equal to 2.
Suppose (cid:22) = (1(cid:22)1 :::h(cid:22)h), where h = h((cid:22)). Proposition 3.1.1 implies that
k;2l l((cid:22))(cid:0)1;2l
(cid:23) (2(cid:22)) (cid:21) (cid:23) (2(cid:22));
and that multiplicityis not trivially zero when ((cid:22);(l((cid:22))(cid:0)1)l(cid:0)j(cid:22)j) really is an increasing
partition: this is equivalent to the inequality
j(cid:22)j+h((cid:22)) h(cid:22)h +(cid:1)(cid:1)(cid:1)+2(cid:22)2 +(cid:22)1 +h
l (cid:21) = ;
l((cid:22))+1 (cid:22)h +(cid:1)(cid:1)(cid:1)+(cid:22)2 +(cid:22)1 +1
which is certainly veri(cid:12)ed when l (cid:21) h((cid:22)). If we apply Newell's identity to the partition
(cid:26) = 2(cid:23), and the integers p = l((cid:22))(cid:0)1 and q = 2l(cid:0)2, where (cid:23) = (1(cid:22)2 :::(h(cid:0)1)(cid:22)h), we get
the equality
l((cid:22))(cid:0)1;2l l((cid:22))(cid:0)1;2l(cid:0)2
(cid:23) (2(cid:22)) = (cid:23) (2(cid:23)):
But h((cid:23)) (cid:20) l(cid:0)1, so that we can apply our induction hypothesis to (cid:23). 2
3.2 Polarization of symmetric functions.
1 k
Let X ;:::;X be a set of k indeterminates. If (cid:25) = ((cid:25)1;:::;(cid:25)h) is some partition, we
(cid:25)
denote by m the monomial symmetric function:
X
(cid:25) j1 (cid:25)1 jh (cid:25)h
m = (X ) (cid:1)(cid:1)(cid:1)(X ) :
j16=(cid:1)(cid:1)(cid:1)=6 jh
When (cid:25) describes the set of partitions of some integer i, of length l((cid:25)) = h (cid:20) k, these
monomial functions provide us with a basis of the space of symmetric polynomials of
j
degree i in the X .
1 1 h h
Now, if P = (p1;:::;p(cid:25)1;:::;p1;:::;p(cid:25)h) is some multiindex of integers between one
(cid:25)
and k, we can polarize m to get a symmetric rational section of L on X, namely
X
(cid:25) (j1) (j1) (jh) (jh)
mP = xp11 (cid:1)(cid:1)(cid:1)xp1(cid:25)1 (cid:1)(cid:1)(cid:1)xph1 (cid:1)(cid:1)(cid:1)xph(cid:25)h (cid:2)(cid:16);
j16=(cid:1)(cid:1)(cid:1)=6 jh
which is regular if and only if h((cid:25)) (cid:20) l. Recall that (cid:16) is the local generator of L we
(cid:25) k l
de(cid:12)ned in 2.2. These mP are clearly independent in S (S V), when the multiindex P is
i i
consideredmodulosymmetriesbetweenindicesp1;:::;p(cid:25)i for eachi, and groups of indices
i i j j
(p1;:::;p(cid:25)i) and (p1;:::;p(cid:25)j) for (cid:25)i = (cid:25)j (we shall say "symmetries of (cid:25)"), and when (cid:25)
describes the set of partitions of i of length at most k, and height at most l: they form
a basis of sections which have a homogeneous polynomial expression of degree i in our
coordinate system.
Gaussian maps and plethysm 10
j j
For each integer j (cid:21) 2, choose complex numbers (cid:11)1;:::;(cid:11)j such that
j j
(cid:11) +(cid:1)(cid:1)(cid:1)+(cid:11) = 0; (1)
1 j
j j j j j
S ((cid:11)) = ((cid:11)1) +(cid:1)(cid:1)(cid:1)+((cid:11)j) 6= 0; (2)
and consider the linear forms
j
X
J j Jm
lj(X) = (cid:11)mX ;
m=1
where J is some multiindexof length j. Then de(cid:12)ne the symmetricfunctions, if (cid:25) is some
partition with no part equal to one,
X
(cid:25) 0 J1 (cid:25)1 Jh (cid:25)h
(cid:28) = l (X) (cid:1)(cid:1)(cid:1)l (X) ;
(cid:25)1 (cid:25)h
J
P
0 1 h i
where the sign J means a sum over multiindices J = (J ;:::;J ), J of length (cid:25)i, such
(cid:25)
that the components of J are pairwise distinct: so that (cid:28) is nonzero only for k (cid:21) i. One
has an expansion
X
(cid:25) (cid:25) (cid:25) (cid:25);(cid:26) (cid:26)
(cid:28) = S ((cid:11))m + t m ;
l((cid:26))>l((cid:25))
where S(cid:25)((cid:11)) = S(cid:25)1((cid:11))(cid:1)(cid:1)(cid:1)S(cid:25)h((cid:11)), from which it is easy to derive the following
Lemma 3.2.1 If k (cid:21) i, when (cid:25) describes the set of partitions of i of length at most k,
(cid:25)
which have no part equal to one, the symmetric functions (cid:28) form a basis, in the space of
1 k
symmetric polynomials of degree i in k indeterminates X ;:::;X , of a supplementary of
1 k
the space of multiples of the sum X +(cid:1)(cid:1)(cid:1)+X .
We shall prove in the following section a polarized analogue of that lemma.
3.3 Symmetric sections vanishing on the diagonal.
(cid:25)
We can polarize the symmetricfunctions (cid:28) exactly as we did for monomialfunctions,
(cid:25)
to get symmetricsections (cid:28)P, where P is a multiindexwhich once again has to be consid-
ered only modulo symmetries of (cid:25). The polarized analogue of the expansion preceeding
lemma 3.2.1 reads as follows:
X X
(cid:25) (cid:25) (cid:25) (cid:25);(cid:26) (cid:26)
(cid:28)P = S ((cid:11))mP + tQ mQ:
l((cid:26))>l((cid:25)) Q
Because of hypothesis (2), we thus obtain a family of independent regular sections of
k l (cid:25)
S (S V), spanning a subspace of the same dimension as the m , when (cid:25) is supposed to
P
have no part equal to one. Note that this dimension is precisely the codimension of the
1 Pk (j)
the subspace of symmetric sections generated by the sums mq = j=1xq , if we restrict
to sections having a homogeneous polynomial expression of degree i in our coordinates.
k
Actually, this is enough to get the i-th jets at point m of all the sections vanishing at
