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FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
KATHRIN KUHNIGK AND LARRY SMITH
KATHRIN KUHNIGK AND LARRY SMITH
KATHRIN KUHNIGK AND LARRY SMITH
KATHRIN KUHNIGK AND LARRY SMITH
KATHRIN KUHNIGK AND LARRY SMITH
# SPRING 1998
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY :
SUMMARY : Let # : G # GL(n, IF) be a representation of a finite group G over the field IF.
Let IF[V ] be the ring of polynomial functions on V = IF n on which G acts via #, and let
IF[V ] G
# IF[V ] be the ring of invariant polynomials. If the characteristic of IF is p and p
divides ïGï, the order of G, then it is known that the transfer map
Tr G : IF[V ] # IF[V ] G
defined by Tr G ( f ) =
P
gÎG
g f has as image an ideal in IF[V ] G of height strictly less than n. In
[1] M. Feshbach described the radical of the image of the transfer. In this note we reprove this
result and in addition describe the variety defined by the extended ideal Im(Tr G )) e in IF[V ].
It turns out that this variety is nothing but the union of the fixed subspaces of the elements
in G of order p.
MATHEMATICS SUBJECT CLASSIFICATION : 13A50 Invariant Theory
Typeset by LS T E X
Let # : G # GL(n, IF) be a representation of a finite group G over the field IF, and let IF[V ]
be the algebra of polynomial functions on the vector space V = IF n . The group G acts on V
and hence also on IF[V ] via the representation # and we denote by IF[V ] G the subalgebra of
Ginvariant polynimials in IF[V ]. (We refer to [5] for basic facts about the invariant theory of
finite groups.) The transfer homomorphism
Tr G : IF[V ] # IF[V ] G
is defined by
Tr G ( f ) = X gÎG
g f , " f ÎIF[V ].
If the characteristic of IF is relatively prime to the order of G then the transfer map is
surjective. By contrast, if the characteristic of IF divides the order of G the image of the
transfer, Im(Tr G ), is an ideal of height strictly less than n [1], [3] [7]. We begin this note
by reproving this result as well as M. Feshbach's unpublished description of the radical of
Im(Tr G ). We then go on to describe the variety defined by the extended ideal (Im(Tr G )) e
#
IF[V ]. It turns out that this variety has a particularly elegant description, namely it is the
union of the fixed point sets of the elements of order p in G, where p is the characteristic
of IF. As an application we show that the Dickson polynomial of least degree d n,n-1 is
always a nonzero divisor in the quotient ring IF[V ] / Im(Tr G ), so this quotient ring has positive
homological codimension (or depth).
§1. Feshbach's Transfer Theorem
Let g ÎGL(n, IF) and define the element # g = 1 - g ÎMat n,n (IF), the algebra of n × n matrices
over IF. The element # g acts on the algebra IF[V ] as a linear twisted differential, i.e.
# g ( f + h) = # g ( f ) + # g (h)
# g ( f · h) = # g ( f ) · h + ( g f ) · #(h) ,
as the following computation shows:
# g ( f + h) = f + h - g f - gh = f - g f + h - gh = # g ( f ) + # g (h)
# g ( f · h) = f · h - g( f · h) = f · h - ( g f ) · h + ( g f ) · h - ( g f ) · ( gh)
= ( f - g f ) · h + ( g f )(h - gh) = # g ( f ) · h + ( g f ) · # g (h).
Let I g # IF[V ] denote the ideal generated by # g (V * ), where as usual V * is the space of linear
forms on V = IF n .
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1
LEMMA 1.1 : Let g ÎGL(n, IF) and Im(# g ) the image of the action of # g on IF[V ]. Then
Im(# g ) # I g .
PROOF PROOF PROOF PROOF PROOF : Choose a basis z 1 , . . . , z n for V * and identify IF[V ] with IF[z 1 , . . . , z n ] as usual.
Both Im(# g ) and I g are graded subsets of IF[z 1 , . . . , z n ] and they agree in grading 1 by
definition. Therefore we may proceed inductively, and suppose that, Im(# g ) and I g agree in
degree < d. If f ÎIF[z 1 , . . . , z n ] is a form, i.e., a homogeneous polynomial, of degree d then
it is a sum of monomials z E = z e 1
1 · · · z e n
n , and so by linearity of # g it is enough to show that
KATHRIN KUHNIGK AND LARRY SMITH
# g (z E ) ÎI g . Without loss of generality we may suppose that e 1 > 0 so we may write z E = z 1 z F
where F = (e 1 - 1, e 2 , . . . , e n ). Since the monomial z F has degree d - 1 we know by induction
# g (z F ) ÎI g . By the twisted derivation formula
# g (z E ) = # g (z 1 ) · z F + ( gz 1 ) · # g (z F ).
By definition # g (z 1 ) ÎI g and since I g is an ideal it follows that both terms on the right hand
side of the preceeding equation are in I g completing the induction step, and hence the proof.
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2
PROPOSITION 1.2 (M. Feshbach) : Let IF be a field of characteristic p, u ÎGL(n, IF) an
element of order p, and P < GL(n, IF) the subgroup generated by u. Then Im(Tr P ) # I u and
hence ht( p Im(Tr P )) £ n - dim IF (V u ).
PROOF PROOF PROOF PROOF PROOF : Note that in the group ring IF(P) we have
Tr P = 1 + u + · · · + u p-1 = (1 - u) p-1 = # p-1
u .
Hence Im(Tr P ) = Im(# p-1
u ) # Im(# u ) # I u by lemma 1.1. The ideal I u is a prime ideal in
IF[V ] since it is generated by linear forms. The height of I u is equal to dim IF (# u (V * )) =
n - dim IF (ker(# u : V # V )) and ker(# u : V # V ) = V u . Hence I u has height n - dim IF (V u ).
Since I u # IF[V ] is prime and IF[V ] P
# IF[V ] is a finite extension the ideal IF[V ] P Ç I u is
prime by the lying over theorem. By the going up theorem ht(IF[V ] P Ç I u ) = ht(I u ) and since
Im(Tr P ) # IF[V ] P Ç I u the result follows.
LEMMA 1.3
LEMMA 1.3
LEMMA 1.3
LEMMA 1.3
LEMMA 1.3 : Let # : G # GL(n, IF) be a representation of a finite group G over the field
IF and H £ G a subgroup. Then Im(Tr G ) # Im(Tr H ).
PROOF PROOF PROOF PROOF PROOF : Choose a right transversal g 1 , . . . , g t for H in G. Then G = H g 1 # · · · # H g t
and hence for any f ÎIF[V ]
Tr G ( f ) = X gÎG
g f = X hÎH
t
X i=1
h g i f = X
hÎH
h(g 1 f + · · · + g t f ) = Tr H ( g 1 f + · · · + g t f )
and the result follows.
THEOREM 1.4
THEOREM 1.4
THEOREM 1.4
THEOREM 1.4
THEOREM 1.4 (M. Feshbach) : Let # : G # GL(n, IF) be a representation of a finite group
over the field IF. Then
(i) Im(Tr G ) # T
ïgï=p
I g , where the intersection runs over all the elements of order p in G,
(ii) ht( p Im(Tr G )) £ n - max
ïgï=p dim IF (V g ) .
PROOF PROOF PROOF PROOF PROOF : For each element g ÎG of order p we have Im(Tr G ) # Im(T r #g# ) by 1.3, where
#g# denotes the subgroup of G generated by g, and the result follows from proposition 1.2.
§2. The Transfer Variety
If # : G # GL(n, IF) is a representation of a finite group and Im(Tr G ) # IF[V ] G the transfer
ideal, then we may extend Im(Tr G ) to an ideal in IF[V ] in the usual way. 1 to an ideal in IF[V ]
This extended ideal defines an algebraic set which we propose to study in this section. To be
specific we introduce:
1 Simply take the ideal in IF[V ] which is generated by the elements in Im(Tr G ).
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FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
DEFINITION DEFINITION DEFINITION DEFINITION DEFINITION : Let # : G # GL(n, IF) be a representation of a finite group over the field IF.
The transfer variety, denoted by X G # V is defined by
X G = n x ÎV#Tr G ( f )(x) = 0 " f ÎTot(IF[V ]) o .
N.b. Since X G is an affine variety, we must use all polynomial functions to define it, and not
just homogeneous ones.
LEMMA 2.1
LEMMA 2.1
LEMMA 2.1
LEMMA 2.1
LEMMA 2.1 : Let g ÎGL(n, IF) and V(I g ) # V the affine variety defined by the ideal
I g #Tot(IF[V ]). Then V(I g ) = V g , where V g # V is the fixed point set of the element g acting
on v.
PROOF PROOF PROOF PROOF PROOF : An element x ÎV belongs to V(I g ) if and only if # g #(x) = 0 for every linear form
#. If x ÎV g then
# g #(x) = #(x) - #(g -1 x) = #(x) - #(x) = 0
so x ÎV(I g ). Conversely, since
# g #(x) = #(x) - #(g -1 x) ,
it follows that x ÎV(I g ) if and only if #(x) = #(g -1 x) for every linear form #. Since the linear
forms separate the points in V it follows x = g -1 x and hence x belongs to V g .
The transfer variety is defined by the radical of the ideal in IF[V ] generated by Im(Tr G ), which
in turn is contained in the intersection of the ideals I g , where g ranges over the elements
of order p in G. Passing to varieties turns the inclusion around and the intersection into a
union, so we get:
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2
COROLLARY 2.2: Let # : G # GL(n, IF) be a representation of a finite group over the field
IF. Then
[
ïgï=p
V g
# X G ,
where the union runs over all the elements of order p in G.
We next describe the transfer variety.
LEMMA 2.3
LEMMA 2.3
LEMMA 2.3
LEMMA 2.3
LEMMA 2.3: Let B # V = IF n be a finite subset and b 0 ÎB. Then there exists a polynomial
function h ÎTot(IF[V ]) such that
h(b 0 ) = 1
h(b) = 0 " b ÎB , b #= b 0 .
PROOF PROOF PROOF PROOF PROOF : It is enough to consider the case of a pair x, y ÎV of distinct points, and prove
the existence of a polynomial function h x, y ÎTot(IF[V ]) with the property
h x, y (x) = 1
h x, y ( y) = 0.
For given this the general case is solved by setting
h = Y bÎB
b#=b 0
h b 0 ,b .
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KATHRIN KUHNIGK AND LARRY SMITH
So suppose x #= y ÎV = IF n . Write
x = (x 1 , . . . , x n )
y = ( y 1 , . . . , y n )
and choose i between 1 and n with x i #= y i . The function h x, y : V # IF defined by
h x, y (a 1 , . . . , a n ) = a i - y i
x i - y i
is linear, though not homogeneous, and has the desired property.
LEMMA 2.4
LEMMA 2.4
LEMMA 2.4
LEMMA 2.4
LEMMA 2.4 : Let # : G # GL(n, IF) be a representation of a finite group G over the field
IF and x ÎV . Then for any f ÎIF[V ] G x of positive degree we have
Tr G ( f )(x) = ïG x ïTr G
G x
( f )(x)
where G x £ G is the isotropy group of x.
PROOF PROOF PROOF PROOF PROOF : Choose elements g 1 , . . . , gm ÎG that are simultaneouly a left and right transver
sal for G x in G. This is always possible by K˜ onig's Lemma [10] pp. 12--13. Then g -1
1 , . . . , g -1
m
is also a left transversal for G x in G and
Tr G ( f )(x) = X gÎG
g f (x) =
m
X i=0
X
hÎG x
h g i f (x)
=
m
X i=1
X
hÎG x
f ((h g i ) -1 x) =
m
X i=1
X
hÎG x
f ( g -1
i hx)
=
m
X i=1
( f ( g -1
i x) + · · · + f ( g -1
i x)
########## ïG x ï ##########
)
=
m
X i=1
ïG x ï f ( g -1
i x) = ïG x ï
m
X i=1
g i f (x)
= ïG x ïTr G
G x
( f )(x)
by the definition of the relative transfer.
PROPOSITION 2.5
PROPOSITION 2.5
PROPOSITION 2.5
PROPOSITION 2.5
PROPOSITION 2.5 : Let # : G # GL(n, IF) be a representation of a finite group G over the
field IF of characteristic p. Then a point x ÎV belongs to the transfer variety X G if and only
if p divides ïG x ï.
PROOF PROOF PROOF PROOF PROOF : The conditions of 2.4 are fullfilled, so, if p divides the order of the isotropy group
G x then by lemma 2.4 Tr G ( f )(x) = ïG x ï P m
i=1 g i f (x) = ïG x ïTr G
G x
( f )(x) = 0 so x ÎX G .
On the other hand suppose p does not divide ïG x ï. The orbit B # V of x is a finite set, so by
lemma 2.3 there is a polynomial function h on V such that
h(x) = 1
h( y) = 0 " y ÎB , y #= x .
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FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
Then by lemma 2.4 and the fact that g 1 (x) , . . . , g t (x) are pairwise distinct, with g i (x) = x only
if g i ÎG x , we get
Tr G (h)(x) = ïG x ï
m
X i=1
g i h(x) = ïG x ï
m
X i=1
h(g -1
i x)
= ïG x ïTr G
G x
(h)(x) = ïG x ï 0 @ X bÎB
h(b) 1 A
= ïG x ïh(x) = ïG x ï #= 0 ÎIF
and hence x #= X G .
COROLLARY 2.6
COROLLARY 2.6
COROLLARY 2.6
COROLLARY 2.6
COROLLARY 2.6: Let # : G # GL(n, IF) be a representation of a finite group over the field
IF of characteristic p. Then
X G = [ gÎG
ïgï=p
V g ,
in other words, the transfer variety is the union of the fixed point sets of the elements in G of
order p.
PROOF PROOF PROOF PROOF PROOF : By 2.5 any x ÎX G is fixed by some element g ÎG with ïgï = p and hence X G #
S
ïgï=p
V g . On the other hand, by 2.2
S
ïgï=p
V g
# X G .
By combining 2.5 with our previous work we arrive at Feshbach's main result for the modular
transfer in invariant theory.
THEOREM 2.7
THEOREM 2.7
THEOREM 2.7
THEOREM 2.7
THEOREM 2.7 (M. Feshbach) : Let # : G # GL(n, IF) be a representation of a finite group
over the algebraically closed field IF of characteristic p. Then
q Im(Tr G ) =
0 B B @ \
ïgï=p
gÎG
I g
1 C C A Ç IF[V ] G .
PROOF PROOF PROOF PROOF PROOF : By 2.6 X G =
S
ïgï=p
gÎG
V g . If the ground field is algebraically closed then passing back
to ideals leads to the equality
q (Im(Tr G )) = \
ïgï=p
gÎG
I g
as ideals in IF[V ], where (Im(Tr G )) is the ideal in IF[V ] generated by Im(Tr G ). The ideals I g
are prime since they are generated by linear forms, hence their intersection with IF[V ] G is
also prime, and the result follows.
PROBLEM :
PROBLEM :
PROBLEM :
PROBLEM :
PROBLEM : Is the representation
q Im(Tr G ) = \
ïgï=p
gÎG
p g ,
where p g = I g Ç IF[V ] G , the primary decomposition of p Im(Tr G )?
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KATHRIN KUHNIGK AND LARRY SMITH
§3. The Depth of IF[V ] G / p Im(Tr G )
It is known that p Im(Tr G ) always contains the Dickson polynomial d n,0 of top degree for any
# : G # GL(n, IF). As an application of Feshbach's transfer theorem we obtain the following
complimentary result which provides a universal example of an invariant not in the image of
the transfer.
PROPOSITION 3.1
PROPOSITION 3.1
PROPOSITION 3.1
PROPOSITION 3.1
PROPOSITION 3.1 : Let # : G # GL(n, IF) be a representation of a finite group over the
field IF of characteristic p. If p divides ïGï then the Dickson polynomial d n,n-1 does not
belong to p Im(Tr G ).
PROOF PROOF PROOF PROOF PROOF : The ideal p Im(Tr G ) # IF[V ] G is invariant under the action of the Steenrod
algebra P * [6]. The action of the Steenrod reduced powers is given by the formulae [9]
P q i-1
(d n,i ) = d n,i-1
for i = n -1, n -2 · · · 1. Therefore, if d n,n-1 Î p Im(Tr G ) then so are all the Dickson polynomials.
Hence the ideal Im(Tr G ) would have height n contrary to feshbach's transfer theorem.
The quotient ring IF[V ] G / Im(Tr G ) is of positive Krull dimension whenever the characteristic
p of IF divides the order ïGï of G. It therefore makes sense to ask for the existence of regular
elements 2 in the quotient ring IF[V ] / Im(Tr G ). We conjecture that the Dickson polynomial
d n,n-1 is such an element in all cases.
2 An element is called regular if it is a nonzero divisor.
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FESHBACH'S TRANSFER THEOREM AND APPLICATIONS
References
[1] M. Feshbach, The Image of the Trace in the Ring of Invariants, Preprint, University of
Minnesota, 1981.
[2] M. Feshbach, pSubgroups of Compact Lie Groups and Torsion of Infinite Height in
H * (BG; IF p ) II, Mich. Math. J. 29 (1982), 299--306.
[3] K. Kuhnigk, DerTransfer in der modularen Invariantentheorie, Diplomarbeit, G˜ ottingen,
1998.
[4] R.J.Shank and D.L. Wehlau, The Transfer in Modular Invariant Theory, Preprint,
Queen's University, Kingston, Ontaria, Canada, 1997.
[5] L. Smith, Polynomial Invariants of Finite Groups, A.K. Peters Ltd., Wellesley, MA,
1995.
[6] L. Smith, P * Invariant Ideals in Rings of Invariants, Forum. Math. 8 (1996), 319--342.
[7] L. Smith, Polynomial Invariants of Finite Groups, A Survey of Recent Developments,
Bull. of the Amer. Math. Soc. 34 (1997), 211--250.
[8] L. Smith, Homological Codimension of Modular Rings of Invariants and the Koszul
Complex, Preprint, AGInvariantentheorie, 1997.
[9] L. Smith and R. M. Switzer, Realizability and Nonrealizability of Dickson Algebras as
Cohomology Rings, Proc. of the Amer. Math. Soc. 89 (1983), 303--313.
[10] H. J. Zassenhaus, Theory of Groups, second edition Chelsea, NY 1958.
Kathrin Kuhnigk Larry Smith
# #
Mathematisches Institut der Universit ˜ at Mathematisches Institut der Universit ˜
at
D 37073 G˜ ottingen D 37073 G˜ ottingen
Federal republic of Germany Federal Republic of germany
email: KUHNIGK@CFGAUSS.UNIMATH.GWDG.DE email: LARRY@SUNRISE.UNIMATH.GWDG.DE
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