The Transfer in the Invariant Theory of
The Transfer in the Invariant Theory of
The Transfer in the Invariant Theory of
The Transfer in the Invariant Theory of
The Transfer in the Invariant Theory of
Modular Permutation Representations II
Modular Permutation Representations II
Modular Permutation Representations II
Modular Permutation Representations II
Modular Permutation Representations II
Mara D. Neusel
Mara D. Neusel
Mara D. Neusel
Mara D. Neusel
Mara D. Neusel
CANADIAN MATHEMATICAL BULLETIN
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AMS CLASSIFICATION :
AMS CLASSIFICATION :
AMS CLASSIFICATION :
AMS CLASSIFICATION : 13A50 Invariant Theory
KEY WORDS :
KEY WORDS :
KEY WORDS :
KEY WORDS :
KEY WORDS : Polynomial Invariants of Finite Groups, Permutation Representation, Transfer
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SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : In this note we show that the image of the transfer for permutation representations
of finite groups is generated by the transfers of special monomials. This leads to a description
of the image of the transfer of the alternating groups. We also determine the height of these
ideals.

Let F be a finite field of char(F) = p. Let N : G  GL(n, F) be a faithful representation of a
finite group G. The group G acts via N on the n­dimensional vector space V =F n . This induces
an action of G on the ring of polynomial functions F[x 1 , . . . , x n ] = F[V ], where x 1 , . . . , x n is
the standard dual basis of V * , via
g f (v) := f (N(g) -1 v) " g V G, f V F[x 1 , . . . , x n ], v V V .
Denote by F[V ] G the ring of polynomials invariant under the G­action, see [10] or [12] for an
introduction to invariant theory of finite groups. The transfer
Tr G : F[V ] & F[V ] G ; f ) X gVG
g f
is a F[V ] G ­module homomorphism. It is surjective if and only if the characteristic of the
ground field F does not divide the group order, i.e., in the non­modular case, where it provides
a tool for constructing the ring of invariants F[V ] G , see Section 2.4 in [12]. On the other hand,
in the modular case, where p P ïGï, the transfer is never zero nor surjective, see Section 11.5
in [12]. This makes the transfer, resp., its image, an interesting object of study and inspired
quite a number of (recent) research, e.g., [2], [3], [6], [7], [8] and [11]. In this note we pursue
the investigation of the image of the transfer of modular permutation representations started
in [8].
§1. A Generating Set
In this section we show that the image of the transfer for a permutation representation is
generated by the transfers of special monomials ­ a result inspired by [4], as it is reworked by
[5].
Let N : GGL(n, F) be a modular permutation representation of a finite group G permuting
a basis x 1 , . . . , x n for the dual vector space V * . Denote by
x E = x E 1
1 · · · x E n
n
V F[x 1 , . . . , x n ]
a monomial with multiindex E = (E 1 , . . . , E n ). We associate to E a partition
l(E) = l 1 (E) , . . . , l n (E)


where for every i = 1 , . . . , n there exists a j V {1 , . . . , n} such that
l i (E) = E s(j) ,
for some sVS n , i.e., the partition l(E) is obtained from the exponent sequence E by reordering
so that it is weakly decreasing. Call a monomial special if
l n (E) = 0 and l i (E) - l i+1 (E) £ 1 " i = 1 , . . . , n - 1.
THEOREM 1.1
THEOREM 1.1
THEOREM 1.1
THEOREM 1.1
THEOREM 1.1: Let N : G  GL(n,F) be a permutation representation of a finite group G.
Then the image of the transfer is generated by
Tr G (x E ) for special monomials x E V F[V ].

MARA D. NEUSEL
PROOF PROOF PROOF PROOF PROOF: Denote by I H F[V ] G the ideal generated by the transfers of special monomials.
Let x E V F[V ] be a non­special monomial. We have to show that
Tr G (x E ) V I .
We assume that Tr G (x E ) Q= 0, for otherwise there is nothing to show. Denote the associated
partition by l(E) = (l 1 (E) , . . . , l n (E)). If l n (E) Q= 0, then
x E = x E  e n ,
where e i denotes the i­th elementary symmetric function, i = 1 , . . . , n, and, E  i = E i - 1. We
have
Tr G (x E ) = Tr G (x E  )e n .
Since the elementary symmetric functions are present in any ring of permutation invariants
it is enough to show that
Tr G (x E  ) V I .
So, without loss of generality, we assume that l n (E) = 0. We want to proceed by induction and
choose for that the dominance order on monomials, i.e.,
x E £ dom x F Û
8 > > < > > :
l 1 (E) £ l 1 (F) and
l 1 (E) + l 2 (E) £ l 1 (F) + l 2 (F) and
. . . and
l 1 (E) + · · · + l n (E) £ l 1 (F) + · · · + l n (F).
Assume to the contrary that I 5Im(Tr G ), and let x E be minimal with respect to the dominance
ordering such that
Tr G (x E ) QV I .
Since x E is non­special, somewhere in the associated partition l(E) is a gap. Let t be the
index of the first occurence of a gap
t = min n il i (E) - l i+1 (E) > 1 o ,
and define the reduced monomial x ”
E to be the one where ”
E is obtained from E by lowering
the largest t of the exponents E i by 1. The reduced monomial x ”
E is, by construction, strictly
smaller in dominance order
x ”
E < dom x E .
Write
Tr G (x E ) = Tr G (x ”
E )e t - R ,
for some polynomial R. Note that
R = Tr G (x ”
E )e t - Tr G (x E ) V Im(Tr G )
shows that R is in the image of the transfer of G. By minimality of x E we have that
Tr G (x ”
E ) V I .
The monomials occuring in R are strictly less in the dominance ordering than x E . This is
shown in the lemma below. Hence R is, by induction, also contained in I. Therefore
Tr G (x E ) = Tr G (x ”
E )e t - R V I ,
what contradicts our assumption and we are done. Y
The following lemma is a revised version of Lemma 10 in [5].
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TRENTE ANS APR ‘
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LEMMA 1.2
LEMMA 1.2
LEMMA 1.2
LEMMA 1.2
LEMMA 1.2: Every monomial x F occuring in
Tr G (x ”
E )e t  = Tr G (x E ) + R 
is smaller with respect to the dominance ordering than x E , and equal if and only if x F is a
term in Tr G (x E ).
PROOF PROOF PROOF PROOF PROOF: Write
x F = x ”
F x j 1
· · · x j t
where
l( ”
E) = l( ”
F) and J := {j 1 , . . . , j t } H {1 , . . . , n}.
We have
l( ”
E) = l 1 (E) - 1 , . . . , l t (E) - 1, l t+1 (E) , . . . , l n (E) 
 .
The partition l(F) associated to F is obtained from l( ”
F) by adding one to some exponents
according to the elements of J (and possibly reordering)
l i (F) =  l s(i) (E) - 1 + 1 J (i) for i = 1 , . . . , t
l s(i) (E) + 1 J (i) for i = t + 1 , . . . , n,
where 1 J is the function taking value one on J and zero elsewhere, and s is a certain permu­
tation corresponding to the possible reordering. Note that this element s V S n permutes the
l i (F) before and after t separately. We need to show that
j
X i=1
l i (F) £
j
X i=1
l i (E)
for j = 1 , . . . , n. For j £ t we get
j
X i=1
l i (F) =
j
X i=1
l s(i) (E) - 1 + 1 J (i) 
 £ 0 @
j
X i=1
l i (E) 1 A - j + ïJ Ç {1 , . . . , j}ï £
j
X i=1
l i (E) ,
where the penultimate inequalitiy follows because reordering lowers the sum of the l i .
Secondly, if j > t then
j
X i=1
l i (F) =
t
X i=1
l s(i) (E) - 1 + 1 J (i) 
 +
j
X
i=t+1
l s(i) (E) + 1 J (i) 

=
t
X i=1
l i (E) +
j
X
i=t+1
l s(i) (E) - t + ïJ Ç {1 , . . . , j}ï £
j
X i=1
l i (E).
Finally, 1
x F = dom x E Û l(F) = l(E).
If x F is a term in the transfer of x E , then x F and x E differ only by a permutation. Therefore
l(F) = l(E).
Conversely, assume that l(F) = l(E). Note that
e t x ”
E = x E + f ,
1 = dom means, of course, that both inequalities hold.
3

MARA D. NEUSEL
for some polynomial f . Without loss of generality we assume that
x 1 · · · x t x ”
E = x E .
The only term in e t x ”
E , whose exponent sequence has the same partition l(E) as x E is
x 1 · · · x t x ”
E (= x E ), because all other terms have a gap at a different index. Moreover, by
construction, x ”
F is a term in the transfer of x ”
E , i.e., there exists an element g V G such that
gx ”
E = x ”
F .
Hence
g(x E ) + g(f ) = g(e t x ”
E ) = e t x ”
F = x F + other terms.
Since l(F)=l(E) and the partitions of the exponent sequences of all other terms involved (i.e.,
the terms of g(f ) and the terms occurring in other terms) are different from these, it follows
that
x F = g(x E )
as claimed. Y
REMARK REMARK REMARK REMARK REMARK: We could derive the preceding result also in the following way: By [2] the ring of
polynomials F[V ] is generated by special monomials as a module over F[V ] S n , and, a fortiori,
as a module over F[V ] G for any permutation group G. Hence in the modular situation the
image of the transfer is given by applying the transfer to these module generators.
Since special monomials have at most degree n(n-1)
2 , we have proved the following.
COROLLARY 1.3
COROLLARY 1.3
COROLLARY 1.3
COROLLARY 1.3
COROLLARY 1.3: The image of the transfer of a permutation representation is generated
by the polynomials of degree at most n(n-1)
2
.
It is worthwhile noting, that the statement of this corollary can be obtained independently of
the preceding theorem, [13]:
PROOF PROOF PROOF PROOF PROOF: The ring generated by the elementary symmetric functions is present in every
invariant ring of a permutation group G, i.e.,
F[V ] S n = F[e 1 , . . . , e n ]  F[V ] G  F[V ] ,
where S n denotes the symmetric group in n letters. The elementary symmetric functions
form a homogeneous system of parameters for F[V ]. Since the ring of polynomials F[V ] is
Cohen­Macaulay, it is free finitely generated over F[e 1 , . . . , e n ]. Hence the maximal degree
of a module generator of F[V ] over F[e 1 , . . . , e n ] can be obtained by dividing the respective
Poincar’ e series. The degree n
2 
 of this polynomial
P F[V ], t 

P F[V ] S n , t 
 = Q n
i=1 (1 - t i )
(1 - t) n =
n-1
Y i=1
(1 + t + · · · + t i )
is the maximal degree of a module generator. Since the transfer Tr G : F[V ] & F[V ] G is an
F[V ] G ­module homomorphism, it is, a fortiori, an F[V ] S n ­module homomorphism. It follows
that Im(Tr G ) is generated as an F[V ] G ­module, i.e., as an ideal, by polynomials of degree at
most n
2 
 . Y
Finally we calculate the height of the image of the transfer.
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TRENTE ANS APR ‘
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THEOREM 1.4
THEOREM 1.4
THEOREM 1.4
THEOREM 1.4
THEOREM 1.4: Let N : G  GL(n,F) be a permutation representation of a finite group G.
Let the characteristic p of the groundfield F divide the group order of G. Then the height
ht(Im(Tr G )) is divisible by p - 1. Moreover we have the following (in­)equalities:
n
p
(p - 1) ³ ht(Im(Tr G ))
= min{k(p - 1)g V G , ïgï = p , g is a product of k p - cycles} ³ p - 1.
PROOF PROOF PROOF PROOF PROOF: By M. Feshbach's transfer theorem, [10] Theorem 6.4.7, the transfer variety
q Im(Tr G ) =
0 B @ \
ïgï=p,gVG
I g
1 C A Ç F[V ] G ,
where I g H F[V ] is the ideal generated by the image of
1 - g : V * &  V * ,
g V G of order p. The ideals I g are generated by linear forms, and therefore prime. By the
Krull relations, the height is preserved when contracting an ideal I g to I g Ç F[V ] G . Hence
ht(Im(Tr G )) = min{ht (I g )g V G, ïgï = p} = min{n - dim(V g )g V G , ïgï = p}.
An element g V G is a product of p­cycles, and hence
dim(V g ) = k + (n - kp) ,
where k denotes the number of p­cycles. Since we assume that p P ïGï, we have that k ³ 1.
Therefore,
n
p (p - 1) ³ ht(Im(Tr G ))
= min{k(p - 1)g V G , ïgï = p , g is a product of k p - cycles}
³ p - 1.Y
§2. The Alternating Group
We apply our results to find the image of the transfer of the alternating groups A n . First recall
from [1] and Section 1.3 [12], or Section 14.2 in [9] that
F[V ] A n = F[e 1 , . . . , e n ,
’ n ]  (r) ,
where
’ n = 8 < :
D n =
Q i<j
(x i - x j ) for p odd (i.e., the discriminant),
Tr A n (x 1 x 2
2 · · · x n-1
n-1 ) for p = 2,
and r is quadratic in
’ n .
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MARA D. NEUSEL
CASE CASE CASE CASE CASE p Q= 2 : If the characteristic p of F is not 2, then the index of A n in S n is prime to p,
so Theorem 5.1 in [11] gives
Im(Tr A n ) Ç F[V ] S n = Im(Tr S n ) ,
and hence we have
 Im(Tr S n )  e
H Im(Tr A n ) F F[V ] A n ,
where (-) e denotes the extended ideal. The discriminant D n is a sum of orbit sums of mono­
mials x E with partition
l(E) = {0 , . . . , n - 1}.
Any such monomial has trivial isotropy group, i.e., the orbit of any such monomial has maxi­
mal length. This in turn means that
D n = Tr A n (x 1 x 2
2 · · · x n-1
n-1 ) - Tr A n (x 2 x 2
1 x 3
3 · · · x n-1
n-1 ) V Im(Tr A n ) ,
is the image of the transfer of a sum of special monomials of highest degree. So we have
  Im(Tr S n )  e
, D n  H Im(Tr A n ).
We claim that these two ideals are equal. To this end take a polynomial f V F[V ] such that
Tr A n (f ) Q= 0. Then
Tr A n (f ) = f 1 D n + f 0
for suitable polynomials f 0 , f 1 VF[V ] S n , because F[V ] A n is, as a module over F[V ] S n , generated
by 1 and D n . So,
Tr S n (f ) = Tr S n
A n
Tr A n (f ) = f 1 Tr S n
A n
(D n ) + 2f 0 .
Hence
f 0 V Im(Tr S n ) ,
and therefore
Tr A n (f ) = f 1 D n + f 0 V   Im(Tr S n )  e
, D n  .
Finally note that the alternating group A n contains a p­cycle, whenever p PïA n ï, p ³ 3. There­
fore by applying Theorem 1.4 we find that
ht(Im(Tr A n )) = min{ht (I g )g V A n , ïgï = p} = p - 1.
REMARK REMARK REMARK REMARK REMARK: Note that this description is valid in both, the non­modular and the modular
situation (except, of course, the calculation of the height: this is always maximal in the non­
modular case).
REMARK REMARK REMARK REMARK REMARK: Consider the modular case, i.e., p £ n. Then the image of the transfer of the full
symmetric group can be found in Theorem 9.18 of [2]: Denote by x E V F[V ] a monomial with
E 1 ³ · · · ³ E n . Rewrite this monomial as
x E = x E 1
1 · · · x E n
n = x 1 · · · x e k 
 k x e k +1 · · · x e k +e k-1 
 k-1 · · · x e k +···+e 1 +1 · · · x n 
 0 ,
and e 0 = n - (e 1 + · · · + e k ). Then Im(Tr S n ) is generated by Tr S n (x E ), where x E satisfies
(1) e i < p for i = 0 , . . . , k, and
(2) e i + e i+1 ³ p for i = 0 , . . . , k - 1.
Note that this is a set of special monomials.
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TRENTE ANS APR ‘
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CASE CASE CASE CASE CASE p = 2 : Let x E = x E 1
1 · · · x E n
n be a special monomial. Denote, similarly as above, by
e i the number of exponents equal to i. This means we can rewrite
x E = x E 1
1 · · · x E n
n =  x j 1
· · · x j e k  k
 x j e k +1
· · · x j e k +e k-1  k-1
· · ·  x j e k +···+e 1 +1
· · · x j n  0
,
where {j 1 , . . . , j n } = {1 , . . . , n}. Note that e 0 + · · · + e k = n. We have that the image of the
transfer of a monomial x E is trivial if, and only if,
ïIso A n
(x E )ï º 0 MOD MOD MOD MOD MOD 2 ,
where Iso A n
(x E ) H A n denotes the isotropy subgroup of x E in A n . Therefore, in order to con­
clude that Tr A n (x E ) = 0, it is enough to find an element of order 2 in Iso A n
(x E ):
If one of the e i 's, say e k , is greater than 3, then
(j 1 j 2 )(j 3 j 4 ) V Iso A n
(x E )
is an element of order 2 in the isotropy group of x E , and Tr A n (x E ) = 0. This leaves to consider
monomials x E such that e 0 , . . . , e k £ 3. If two (or more) of the e i 's, say e k and e k-1 , are greater
than one, then we find
(j 1 j 2 )(j e k +1 j e k +2 ) V Iso A n
(x E ).
Again, we can conclude that Tr A n (x E ) = 0. Hence, the image of the transfer of the alternating
group A n is generated by the images Tr A n (x E ) of special monomials x E such that
{E 1 , . . . , E n } = 8 < :
{0 , . . . , n - 1} or
{0 , . . . , n - 2} or
{0 , . . . , n - 3} ande i = 3 for some i = 0 , . . . , k.
Note that we find precisely two monomials of the first type with different orbits, namely
’ n = Tr A n
 x 1 x 2
2 · · · x n-1
n-1  ,
and
’ 
n = Tr A n
 x 2 x 2
1 x 3
3 · · · x n-1
n-1  = 0 @ Y i<j
(x i + x j ) 1 A +
’ n = D n +
’ n ,
where the discriminant
D n = Y i<j
(x i + x j ) = Tr S n
 x 1 x 2
2 · · · x n-1
n-1 
generates the image of the transfer of S n , see [6].
Observe that for n = 3 (i.e., the non­modular situation) the above given set of monomials gives
Tr A 3 (x 2
1 x 2 ) =
’ 3 , Tr A 3 (x 2
1 x 3 ) =
’ 
3 = e 1 e 2 + e 3 +
’ 3 , Tr A 3 (x 1 x 2 ) = e 2 , and Tr A 3 (x 1 ) = e 1 ,
and we once more see that the transfer is surjective.
Finally, we can apply, as in the case of odd characteristic, Theorem 1.4 and find that the height
ht(Im(Tr A n )) = 2 for n ³ 4, i.e, for the modular situation. However, we could also prove this,
without making use of M. Feshbach's transfer theorem, by direct calculation: Consider the
subgroup
Z/2 < A n ,
generated by (12)(34) V A n . By Lemma 1.3 of [7] we have
 Im(Tr A n )  e
H Im(Tr Z/2 ) H F[V ] Z/2 .
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MARA D. NEUSEL
Hence, we have, by going­up and down, that
ht  Im(Tr A n )  £ ht  Im(Tr Z/2 )  .
By Theorem 2.4 in [8]
ht  Im(Tr Z/2 )  £ 2.
Therefore, also the image of the transfer of A n has height at most 2. We claim that its height
is precisely 2. Since the image of the transfer is non­trivial, its height is positive, and we
assume to the contrary, that
ht  Im(Tr A n )  = 1.
Then also the ideals
Im(Tr A n ) Ç F[V ] S n I Im(Tr S n )
have height 1. Recall that the image of the transfer of the symmetric group S n is a prime ideal
generated by the discriminant, see [6]. This implies
Im(Tr A n ) Ç F[V ] S n = Im(Tr S n ).
However, take
’ n ,
’ 
n
V Im(Tr A n ).
Then the orbit of, say,
’ n under S n is {’ n ,
’  n }, so
D n =
’ n +
’  n ,
’ n ·
’  n
V F[V ] S n .
The second polynomial is not in the image of the transfer of S n
’ n ·
’  n
QV 
’ n +
’  n  = (D n ) = Im(Tr S n ) = F[V ] S n ,
because
’ n ·
’  n V F[V ] S n is irreducible. Since
’ n ·
’ 
n
V Im(Tr A n ) Ç F[V ] S n ,
this is a contradiction. So, ht  Im(Tr A n )  = 2 for n ³ 4.
We summarize the results of this section in a theorem.
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1
THEOREM 2.1: We assume that the characteristic divides the order of the alternating
group, for otherwise the transfer is surjective. Set
x E = x E 1
1 · · · x E n
n =  x j 1
· · · x j e k  k
 x j e k +1
· · · x j e k +e k-1  k-1
· · ·  x j e k +···+e 1 +1
· · · x j n  0
and
e 0 = n - (e 1 + · · · + e k ) and {j 1 , . . . , j n } = {1 , . . . , n}.
If the characteristic p is odd then the image of the transfer of the alternating group A n is
generated by the discriminant D n and Tr A n (x E ), where
(1) e i < p for i = 0 , . . . , k, and
(2) e i + e i+1 ³ p for i = 0 , . . . , k - 1.
Moreover its height is precisely p - 1.
If the characteristic is even, then the image of the transfer is generated by Tr A n (x E ), where
{E 1 , . . . , E n } = 8 < :
{0 , . . . , n - 1} or
{0 , . . . , n - 2} or
{0 , . . . , n - 3} ande i = 3 for some i = 0 , . . . , k.
Moreover, this ideal has height precisely 2.
References
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TRENTE ANS APR ‘
ES, BIS
[1] Marie­Jos’ e Bertin, Anneau des Invariants du Groupe Altern‘e, en Caract’eristic 2, Bull.
Sc. Math., 2 e S‘ erie 94 (65­72), 1970.
[2] H. E. A. Campbell, I. P. Hughes, R. J. Shank and D. L. Wehlau, Bases for Rings of
Coinvariants, Transformation Groups 1 (1996), 307­336.
[3] Mark Feshbach, p­Subgroups of Compact Lie Groups and Torsion of Infinite Height in
H * (BG; F p ), Mich. Math. J. 29 (1982), 299­306.
[4] Manfred G˜ obel, Computing Bases for Rings of Permutation­Invariant Polynomials, J.
Symbolic Computation 19 (1995), 285­291.
[5] Nelson Killius, G˜obel's Bound for Permutation Groups, unpublished notes, Evanston,
1995.
[6] Kathrin Kuhnigk, Der Transfer in der modularen Invariantentheorie, Diplomarbeit,
G˜ ottingen, 1997.
[7] Kathrin Kuhnigk and Larry Smith, Feshbach's Transfer Theorem and Applications,
preprint, AG Invariantentheorie 1998.
[8] Mara D. Neusel, The Transfer in the Invariant Theory of Modular Permutation Repre­
sentations, Pacific Journal of Mathematics, to appear.
[9] Mara D. Neusel, A Selection from the 1001 Examples. The Atlas of the Invariant Theory
of Finite Groups, in preparation.
[10] Mara D. Neusel and Larry Smith, Invariant Theory of Finite Groups, book manuscript,
to appear.
[11] R. James Shank and David L. Wehlau, The Transfer in Modular Invariant Theory, J. of
Pure and Applied Algebra 142 (1999), 63­78.
[12] Larry Smith, Polynomial Invariants of Finite Groups, Research Notes in Mathematics,
A. K. Peters Ltd., Wellesley, MA, 2nd corrected printing 1997.
[13] Larry Smith, electronic communication, October 25, 1999.
Mara D. Neusel
University of Notre Dame
Department of Mathematics
370 CCMB
Notre Dame, IN 46556­5683
USA
neusel.1@nd.edu
9