Table Of ContentTopics in
Cotnputational Algebra
Edited by
G. M. PIACENTINI CATIANEO
and
E. STRICKLAND
Department of Mathematics,
University ofR ome Il 'Tor Vergata', ltaly
Reprinted from Acta Applicandae Mathematicae,
Volume 21, Nos. 1 & 2 (1990)
SPRINGER SCIENCE+BUSINESS MEDIA, B.V.
Library of Congress Cataloging in Publication Data
Topies in eomputational algebra / edited by G.M. Piaeentini
Cattaneo, E. Striekland
p. cm.
"Papers presented at the Computational Algebra Seminar held at the
University of Rome 'Tor Vergata,' 9-11 May 1990"--P.
"Acta appllcandae mathematicae, volume 21, nas. 1 and 2,
October/November 1990"--P.
rSBN 978-94-0l0-5514-7 ISBN 978-94-011-3424-8 (eBook)
DOI 10.1007/978-94-011-3424-8
1. Algebra--Data processing--Congresses. r. Piaeentini Cattaneo, G. M.
II. Strickland, E. (Elisabetta), 1948- III. Computational
Algebra Seminar (1990 University of Rome "Tor Vergata")
QAI55.7.E4T67 1990
512' .0028--dc20 90-27032
ISBN 978-94-010-5514-7
Printed an acid-free paper
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© 1990 by Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1990
Softcover reprint of the hardcover 1s t edition 1990
No part of the material protected by this copyright notice may be reproduced or utilized in any fonn
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Contents
G. M. PIACENTINI CATTANEO and E. STRICKLAND / Foreword
V. G. KAC and M. WAKIMOTO / Branching Functions for Winding Subalgebras
and Tensor Products 3
H. P AHLINGS / Computing with Characters of Finite Groups 41
F. CELLER, J. NEUBUSER, and C. R. B. WRIGHT / Some Remarks on the
Computation of Complements and Normalizers in Soluble Groups 57
MIKE STILLMAN / Methods for Computing in Algebraic Geometry and Com-
mutative Algebra 77
JEFFREY B. REMMEL / Combinatorial Algorithms for the Expansion of Various
Products of Schur Functions 105
VESSELIN DRENSKY / Polynomial Identities for 2 x 2 Matrices 137
NEIL L. WHITE / Cayley Factorization and a Straightening Algorithm 163
EDWARD FORMANEK / The Nagata-Higman Theorem 185
ROSA Q. HUANG, GIAN-CARLO ROTA, and JOEL A. STEIN / Supersym-
metric Bracket Algebra and Invariant Theory 193
DAV ID A. BUCHSBAUM / Aspects of Characteristic-Free Representation Theory
of GLI!' and Some Applications to Intertwining Numbers 247
Acta Applicandae Mathematicae 21: 1, 1990.
© 1990 Kluwer Academic Publishers.
Foreword
This volume collects the papers presented by 10 invited speakers at the Semester of
Computational Algebra which was held at the Department of Mathematics of the Second
University of Rome 'Tor Vergata' and was organized by Profs. G.M. Piacentini Cattaneo
and E. Strickland during the period January-May 1990.
The aim of the Semester was to give an update on interesting techniques based on
algorithms in different branches of algebra, with the idea of emphasizing the computa
tional aspects of each branch. For this reason, the speakers were chosen from among the
foremost experts in the various fields. The general fields covered are group theory,
commutative algebra, algebraic geometry, representation theory, ring theory, invariant
theory, and supersymmetric algebra.
The editors believe that the Semester largely succeeded in reaching its intended
objectives and, therefore, wish to thank the speakers and all those who contributed to this
success. They want to express their recognition to the institutions who have contributed
with their financial support: Ministero dell'Universita e della Ricerca Scientifica e
Tecnologica, Department of Mathematics of the Second University of Rome, Centro
Matematico V. Volterra, and the Second University of Rome 'Tor Vergata'. Also, a
special thank you to the Gruppo Nazionale 'Teoria dei Gruppi e Algebra non com
mutativa' of the MURST and to the Progetto Strategico 'Algoritmi connessi allo studio di
strutture algebriche' of the Consiglio Nazionale delle Ricerche for their generous
contributions.
Finally, we wish to record a special thank you to Prof. L. Accardi, Director of the
Centro Matematico V. Volterra, for his cooperation during the various stages of the
organization of the Semester.
The Editors:
G.M. PIACENTINI CATTANEO
E. STRICKLAND
Acta Applicalldae Mathematicae 21: 3-39, 1990, 3
© 1990 KhlWer Academic Publishers.
Branching functions for winding subalgebras and tensor products
V.G. KAC' AND M. WAKIMOTO"
'Department of Math., MIT, Cambridge, MA 02139, USA
"Department of Math., Mie University, Tsu 514, Japan
§o. Introduction.
0.1. One of the basic problems of representation theory is to find a decomposition
of an irreducible representation of a group with respect to a subgroup. Namely, suppose
that we have a representation 71' of a group G in a vector space V and suppose that with
respect to a subgroup 5 this representation decomposes into a direct sum of irreducible
representations:
7r=EBi7ri, V=EBiVi
Given an irreducible representation of 5 one denotes by [71' : the number of repre
(J' (J']
sentations of 5 equivalent to in this decomposition, and calls this number a branching
(J'
coefficient. An important problem is to compute the branching coefficients.
A special case of this problem is the decomposition of a tensor product. In this case
G = 5 X 5, 5 is the diagonal subgroup of G, V = V' is) V", where (V', 71") and (V", 71''')
are some irreducible representations of 5, and the problem is to compute the numbers
[71" is) 71'" : (J'].
0.2. In the present paper we study branching coefficients for positive energy repre
sentations of affine algebras. Let us recall the basic definitions in the "non-twisted" case
(r = 1). See [8] for details.
Let £I be a complex simple finite-dimensional Lie algebra of rank C, and let rjJ(.,.) be
its Killing form. "F±i x a triangular decomposition £I = iL + IJ + "+, where IJ is a Cart an
IJ
sub algebra and are maximal nilpotent subalgebras, and let ()V E be the coroot
corresponding to the highest root. Let hV = rjJ(()V,()V) be the dual Coxeter number and
let (xly) = rjJ( x, y )/2h v be the normalized invariant form on g. The affine algebra g'
associated to £I (called also the affine algebra of type X; 1) , where Xl is the type of g) is
constructed as follows. Let qt, e l] be the algebra of Laurent polynomials in t, and let
us view the loop algebra g[t, ell = qt, t-l] is)c £I as an (infinite-dimensional) Lie algebra
over C. Then
g' = g[t,el] + CK
is the L1ique non-trivial central extension of g[t, ell by a I-dimensional center CK.
Explicitly, it can be defined by the following commutation relations:
[x(m), y(n)] = [x, y](m + n) + mOm,_n(xly)K,
where x(m) E g[t,el] stands for tm is) x (m E 1., x E g). We identify 9 with the
sub algebra 1 is) £I of g', and let ~' = IJ + CJ( be the Cartan subalgebra of g'. Let also
n± = "± + t±lg[t±l]. Then we have tEe triangular decomposition g' = n_ + ~' + n+.
Supported in part by NSF grant DMS-8802489 and Sloan grant 88-10-1.
AMS subject classification (1980). 20C35.
Key words. Affine algebras, branching functions.
4 v. G. KAC AND M. W AKIMOTO
An arbitrary affine algebra is a direct sum of the Lie algebras of the form g' and their
twisted analogues g'(O",s) (see below).
It is often convenient to consider g' as an ideal of co dimension 1 in the Lie algebra
g = g' + Cd, with commutation relations
[d,x(m)] = mx(m), [d,K] = 0,
and let ~ = ~' + Cd be its Cartan subalgebra. One extends the normalized bilinear form
(.1.) from i) to ~ by letting (i)lCK + Cd) = 0, (KIK) = (did) = 0, (KId) = l.
A representation 7r of the affine algebra g' in a vector space V is called a positive
energy representation if a) 7r(K) = kI, k E C, and b) 7r can be extended to the whole g
such that -7r( d) is diagonalizable and its eigenvalues are non-negative integers.
The number k is called the level of V. The eigenspace decomposition
V = EenEZ+ V(n)
with respect to -7r( d) is called the energy decomposition; if v E V(n) we say that v has
energy n.
Since [d,g] = 0, the energy decomposition is g-invariant, and we denote by 1f the
representation of g in V(O). It is easy to show that the map 7r (k,1f) establishes a
f-7
bijective correspondence between the set of (equ°iv alence classes of) positive energy irre
ducible representations (7r, V) such that V(O) i- of g' and the set of pairs (k, 1f), where
k E C and 1f is an irreducible representation of g (considered up to equivalence). (Given
k E C and an irreducible represen°ta tion 1f in V(O), °w e extend 1f to g+ := g[t] + CK + Cd
by letting 1f(K) = kIVr.O)' 1f(d) = and 1f(x(n» = for n > 0, and let 7r be the irreducible
quotient of the inducedrepresentationof g in the space U(g) 0u(9+) V(O). This gives us an
irreducible positive energy representation 7r of g' corresponding to the pair (k, 1f).)
0.3. We shall consider only the irreducible positive energy representations (7r, V) of
g' such that (1f, V(O) is an irreducible highest weight representation of g. In other words,
we shall assume that there exists a non-zero vector vk,X E V(O), where X E i), such that
These representations are parameterized by the pairs (k, A), k E R, X E i)*. It is more
convenient to represent the pair (k, X) by an element A E ~'. such that Ali) = X and
A(K) = k. The corresponding representation 7r>. of g' is denoted by L(A) and is called the
irreducible highest weight representation of g' with highest weight A.
The vector v>. := v X is called the highest weight vector; it is the unique up to a
k
non-zero constant factor ~ector in L(A) satisfying equations
7r>.(n+)v>. = 0, 7r>.(h)v>. = A(h)v>. for h E ~'.
Especially important are the representations L( A) of g' which can be lifted to a (pro
jective) representation of the corresponding loop group. These are called integrable highest
weight representation. They are parameterized by the set
BRANCHING FUNCTIONS FOR WINDING SUBALGEBRAS AND TENSOR PRODUCTS 5
r
Here P + C is the set of highest weights of finite-dimensional irreducible representations
ofg.
The basic tool for study of integrable L('x) is the so called Weyl-Kac character formula
[6] for the function ch>. on Y:= {v E ~IRe(vIK) > O} defined by
Recently a similar character formula has been established for a larger class of the L('x),
those with an "admissible" highest weight [12]. Their "normalized" characters are Jacobi
modular forms and, conjecturally, these representations are characterized by this property.
(These are also the only ones for which the Kazhdan-Lusztig polynomials are trivial.)
In the present paper we consider the best studied (see [13]), principal admissible highest
weight representations. Their levels may be arbitrary rational numbers k (called principal
admissible) such that k + hV ~ hV /u, where u E N is the denominator of k. In the case
when k E Z+ all principal admissible representations are integrable (but all representations
of fractional level are not).
0.4. A natural class of sub algebras of the Lie algebra g' (and similarly of an arbitrary
g
affine algebra) to consider is the following. Let be a reductive subalgebra of g, let 0' be
9
an automorphism of and let sEN be such that O'S = 1. Define an automorphism if of
the sub algebra g[t, e l] + CK of g' by letting
_ 27l'in
O'(x(n)) = (exp -)O'(x)(n), O'(K) = K,
s
and denote by g' (0', s) the fixed point set of if.
0.5. We shall consider only the representations L('x) of g' which are completely re
ducible with respect to g'(O',s). (This is always the case when ,X E P+.) The branching
coefficients of such g'(O',s) in a representation L('x) of g', i.e. the numbers [L(,X): t(Il)],
where t(ll) is an irreducible highest weight representation of g( 0', s), are almost always
zero or infinity. To get around this, let g( 0', s) = "_ + 6+ "+ be the induced triangular
decomposition of g( 0', s), i.e. "± = n± n g( 0', s) and 6' = ~' n g( 0', s), and let
These numbers are always finite, and we can consider the power series
L
b~ = qm>.," ['x: Il]nqn.
nEZ+
The number m>',/l is a rational number called the modular anomaly which is defined as
follows. Let q = e2 ..i Tj then the above series converges to a holomorphic function in T for
1m T > O. According to [10, Proposition 4.36]' provided that ,X E P+, there exists a unique
number m>',/l such that the function b~( T) is a modular function, i.e. is fixed under the
action of a principal congruence subgroup feN), some N. (Explicit formulas for the m>',/l
may be found in [11] and in the paperj of course, these numbers depend also of the g' and
the subalgebra.)
6 V. G. KAC AND M. WAKIMOTO
The functions b~(r), clearly, completely describe the decomposition of L(A) with re
spect to the subalgebra in question. They are called branching functions. If [L(A) : L(fl)] <
00, then, of course, [L(A) : L(fl)] = limdD b~(r). In general, we study the asymptotics of
b~(r) as r ! a instead. Namely, since the b~(r) are modular functions, we have as r ! 0,
i=
provided that b~ 0:
b~( r) ~ a().., fl)e ,;~ 9(>',!'),
where a( A, fl) > a and g( A, fl) ~ a are real numbers called the asymptotic dimension and
growth of the branching function. Here and in the rest of the paper f( r) ~ g( r) means
that limdD f(r)/g(r) = l.
In all known examples the growth depends only on the algebra g', the subalgebra and
the level k of A, but this is an open problem, which we will refer to as the basic conjecture,
even in the case of integrable L( A) and the subalgebra g( 1, 1) (cf. [11]). Incidentally,
the knowledge of the above asymptotics allows one to compute the asymptotics of the
branching coefficients [A : fl]n as n -+ 00 by making use of a Tauberian theorem (see [10]
and [11]).
0.6. An important version of a special case of branching functions are string functions
c~, which correspond to g = I), (J = 1, s = 1, i.e. which describe the multiplicities of
weights of L( A). Namely, given ).., fl E Ij" of level k, we define as usual, the weight space
L(A)!, = {v E L(A)lh(v) = fl(h)v for all h E Ij'}, and let
L
c~ = qm, .• (dimL(A)!, n L(A)(n))qn,
nEZ+
where we let m>. = 1:\ + pl2 /2( k + h V) - Ipl2 /2h V to be the modular anomaly of A (see
below) and m>.,!, = m>. -11T12/2k is the modular anomaly in this case (as usual, p is the
half-sum of positive roots for g). This is a modular form of weight -£/2, and it is related
to the corresponding branching function by the equation
where G( r) is a modular form of weight £/2 given in §2.2.
String functions for integrable L(A) were studied in great detail in [9], [10], [11] (see
also [8, Chapters 12 and 13]). The key result of this work is an explicit transformation
formula for the normalized characters
under the action of the involution S: r -1/r (defined in §4.3). Here we identify
1---+
q with the function e-K (v) = e-(Kiv) on Ij. (This result was extended in [13, Theorem
3.6] to the case of principal admissible L(A); see formula (4.3.1) of the present paper.)
One deduces from this result an explicit transformation formula for string functions under
the involution S [10, Theorem A]. Since the transformation formula together with the
polar parts of the q-expansions completely determine modular forms, this allows one to
compute the string functions explicitly in many interesting cases. Furthermore, it turns
BRANCHING FUNCTIONS FOR WINDING SUBALGEBRAS AND TENSOR PRODUCTS 7
out that the asymptotics of the string functions c~ is independent of ft, which allows one to
find this asymptotics explicitly ([10, Proposition 4.21] or [8, Chapter 13],) proving thereby
the validity of the basic conjecture in this case. (Incidentally, for admissible A the string
functions C~(T) fail to be modular forms [14].)
0.7. General branching functions in the case of the subalgebras of the form g' (1,1) =
g[t,C1] EB CK of g' and integrable L(A) were studied in detail in [11]. Again from the
tranformation formula for the normalized characters one deduces the transformation law
for these branching functions under the involution S [11, Theorem A]. One derives from
this definitive results on asymptotics (which prove the validity of the basic conjecture)
only in the case oftensor products [11, (2.7.15) and §3.4]. For general branching functions
we derive the basic conjecture from the conjectural positivity of certain matrix elements
of the transformations S in the basis of branching functions [11, p. 188].
0.8. In the present paper we consider the subalgebras
of g', called the winding subalgebras. (This is the simplest case different from g'(l, 1).)
The first basic result of the paper is Theorem 2.1 which gives an explicit expression of the
branching functions b~ for winding subalgebras in integrable highest weight representations
L( A) in terms of string functions. The special case of this theorem, when the level of L( A) is
1, is Theorem 2.2, which gives a solution to Frenkel's conjecture [4]. Theorem 2.1 leads also
to Conjecture 2.2 on the asymptotics of the branching functions for winding subalgebras.
We were able to prove only that it holds for all sufficiently large u.
0.9. Next we compare branching functions b~ for winding subalgebras with branching
functions b~0p for tensor products L(A) @ L(ft). It is clear from [13, Corollary 4.1 and
Theorem 3.6] that the branching functions b~0p are modular functions provided that A
is integrable and ft is admissible. This is the case studied in the present paper. As in
the case of winding subalgebras, we find an explicit expression of the b~0p in terms of
string functions (Theorem 3.1). Comparing Theorems 2.1 and 3.1, we see that a branching
function b~ for a winding subalgebra g[n] coincides with a branching function for tensor
product L(A)@L(ft), where ft = ((u-1 -l)hV,O), provided that u is relatively prime to
h v and rV (Proposition 3.2). This (still mysterious) coincidence indicates a remarkable
interplay between the integrable and admissible representations. As in the case of winding
subalgebras, Theorem 3.1 leads to Conjecture 3.1 on asymptotics of the b~0p, generalizing
the known result in the integrable case. After this paper was completed we received preprint
[1] where Theorem 3.1 is derived in the integrable case using a free field resolution.
0.10. The remarkable feature of the theory of integrable, and, more generally, prin
cipal admissible highest weight representations is the SL2(Z)-invariance of the C-span of
normalized characters (in the twisted case, SL2(Z) should be replaced by a slightly smaller
subgroup f; see Proposition 4.3), hence the SL (Z)-invariance of the C-span of branching
2
functions b~0P, where A (resp. ft) runs over all integrable (resp. principal admissible)
weights of fixed level. (A similar result holds for arbitrary subalgebras of the form g' (1,1),
see [11]. This follows from the S-invariance of normalized characters.) Due to the above
coincidence, this is the case also for the branching functions for winding subalgebras g[n]
8 V. G. KAC AND M. WAKIMOTO
provided that u is relatively prime to h v and r v. However, in the general case we have
only the roC u )-invariance (see §4.3). This is a general feature of the subalgebras 9' (0", s).
0.11. The last, Section 4, contains some preparatory material for our forthcoming
paper with E. Frenkel [3]. It deals with functions CP).,p., which are branching functions for
tensor products of the level 1 integrable representations with arbitrary principal admissible
representations (see (4.1.1) and Theorem 4.1). These functions previously appeared in this
context in [12, Proposition 3].
It turns out that the functions CP).,p. can be obtained by a simple limiting procedure
from the characters of the principal admissible representations (Proposition 4.2). As will
be explained in [3] this procedure naturally appears in the quantization of the Drinfeld
Sokolov reduction developed in [2]. As a result, one obtains that the functions CP).,p. are
characters of the so called extended conformal algebras, which are higher rank generaliza
tions of the Virasoro algebra (cf. [15]). In the particular case of 9 = s.e2(C), this procedure
is equivalent to taking the residue of the admissible characters. As was shown previously
in [16], this reproduces the Virasoro characters. It is worth mentioning that the limiting
procedure gives a non-zero result only for "non-degenerate" principal admissible weights;
in particular, the integrable characters always give zero.
The main result of this last section is Theorem 4.4 which give a transformation formula
for the CP).,p. under the action of S, obtained from the limiting procedure (which is simpler
than that obtained from tensor products). This formula will be applied in [3] to calculate
the fusion rules for the extended conformal algebras.
Thus Theorem 4.1b means the coincidence of two theories of extended conformal
algebras at least on the character level in the simply laced or twisted case. (In the case of
B?) they are different, as can be seen by comparing Theorem 2.2' and Proposition 4.2.)
Note that though the set of principal admissible representations of given fractional
level carries quite a few features of a conformal field theory (like modular invariance, the
unique vacuum, the involution), it can't be a conformal field theory since, for example,
its fusion rules computed by Verlinde's formula [17] may be negative. This makes it quite
remarkable that a "reduction" of this theory indeed produces a conformal field theory.
0.12. We would like to thank E. Getzler and M. Hopkins who pointed out that the
study of branching functions for winding subalgebras may be important 'for the theory of
cohomological operations in the elliptic cohomology, which stimulated our research. We
thank E. Frenkel for his patient explanations of his (joint with Feigin) work [2] and for
collaborati~n in Section 4. We thank D. Jerison for consultations on asymptotics.
The first author wishes to thank E. Strickland for the invitation to give a talk in
January 1990 in the Universita di Roma II on the subject of this paper, for her persistence
in having the paper written and for her lavish hospitality. The second author acknowledges
the hospitality of MIT, where during his stay in the spring of 1990 the paper was completed.
§1. Notation and preliminaries.
1.1. Let I = {O, 1, ... ,.e}, .e 2 1. Recall that an affine matrix is a square matrix
A = (aij)i,jEI such that aii = 2, -aij E Z+ for i # j, aij = 0 implies aji = 0, and
there exists a unique sequence (ao, ... ,ai) of positive relatively prime integers, called the
null-lIectorof A, such that (ao, ... ,ai)(fA) = O.
