Lectures on noncommutative symmetric functions [Lecture notes]

49 Pages·1998·0.489 MB·English

by  Jean-Yves Thibon

LECTURES ON
NONCOMMUTATIVE
SYMMETRIC FUNCTIONS
Jean-Yves Thibon
Institut Gaspard Monge
Universit(cid:19)e de Marne-la-Vall(cid:19)ee
Cit(cid:19)e Descartes, 5 Boulevard Descartes
Champs-sur-Marne, 77454 Marne-la-Vall(cid:19)ee cedex
France
[email protected]
July 1998
Abstract
This is the text of lectures delivered at the RIMS (Kyoto University) in July 1998. It
presents the basic structures of the theory of noncommutative symmetric functions, with
emphasis on the parallelwith the commutative theory and on the representation theoreti-
calinterpretations. Someexamplesinvolvingdescentalgebrasandcharactersofsymmetric
groups are discussed in detail.
Introduction
The theory of noncommutative symmetric functions is the outgrowth of a program ini-
tiated in 1993. The starting point was the theory of quasi-determinants of Gelfand and
Retakh [33, 34], which is the analogue of the theory of determinants for matrices with
entries in a noncommutative ring. Many classical determinantal identities can be lifted
to the level of quasi-determinants [58]. Such determinantal identities are widely used in
the theory of symmetric functions, and most of them can be translated into formulas
involving Schur functions. The original idea was then to look for some noncommutative
analogueof the theory of symmetric functions, in which quasi-determinants would replace
determinants.
Such a theory does exist, and the quasi-determinants arise in applications to envelop-
ing algebras, roots of noncommutative polynomials, noncommutative continued fractions,
Pad(cid:19)e aprroximants or orthogonal polynomials [32, 85, 35, 36]. These calculations usually
take place in a skew (cid:12)eld, which is the (cid:12)eld of quotients of a free associative algebra Sym,
which appears as the proper analogue of the classical algebra of symmetric functions.
It is well known that symmetric functions have several interpretations in represen-
tation theory. It turns out that most of these interpretations have an analogue in the
noncommutative case. It is this aspect of the theory which will be the main subject of
these lectures.
After reviewing brie(cid:13)y the relevant features of the classical theory (Section 1), we
describe in detail the algebraic structure of the Hopf algebra of integral noncommutative
symmetric functions (Section 2), including the duality with quasi-symmetric functions,
and the connection with descent algebras. The representation theoretical interpretations
are discussed in Section 3. The (cid:12)rst one, involving Hecke algebras at q = 0, leads us to
the de(cid:12)nition of quantum quasi-symmetric functions. The second one, a quantum matrix
algebra at q = 0, provides us with the relevant analogues of the Robinson-Schensted
correspondence and of the plactic algebra. Finally, a quantized enveloping algebra at
q = 0, for which one has a natural notion of Demazure module, and the character formula
for these modules, leads to a new action of the Hecke algebra on polynomials, from
which one can de(cid:12)ne quasi-symmetric and noncommutative analogues of Hall-Littlewood
functions. Section 4 presents a choice of examples. First, we analyze three idempotents of
the group algebra of the symmetric groups involved in the combinatorics of the Hausdor(cid:11)
series, and exhibit a natural one-parameter family interpolating between them. Next,
we show that similar calculations can give the decomposition of the tensor products of
certain representations of symmetric groups. We conclude by the diagonalization of the
iterated left q-bracketing operator of the free associative algebra.
Thesenotescorrespondtoaseriesoflecturesdeliveredattheworkshop\Combinatorics
and Representation Theory" held at the Research Institute for Mathematical Sciences
(Kyoto University) in July 1998. I would like to thank the organizers for their invitation.
Notations. | We essentially use the notations of Macdonald's book [78] for commu-
tative symmetric functions. A minor change is that the algebra of symmetric functions
is denoted by Sym, the coe(cid:14)cients being taken in some (cid:12)eld K rather than in Z. The
symmetric group is denoted by Sn. The Coxteter generators (i;i+1) are denoted by si.
1
1 Some highlights of the commutative theory
Modern textbooks in Algebra usually close their account of symmetric functions with
the so-called \Fundamental Theorem" of the theory, stating that the ring of symmetric
polynomials in n variables
Sn
Sym(X) = K[x1;:::;xn]
(K is some (cid:12)eld of characteristic 0) is freely generated by the elementary symmetric
polynomials
X
ek = xi1xi2 (cid:1)(cid:1)(cid:1)xik (k = 1;2;:::;n):
i1<:::<ik
Thus, Sym(X)isjustapolynomialalgebraK[e1;:::;en], withaparticulargradingde(cid:12)ned
by deg(ek) = k.
This is not, however, the end of the story, and the 475 pages of the second edition of
Macdonald's book [78] do not su(cid:14)ce to exhaust the subject, which is still an active area
of research.
The algebra Sym of symmetric functions is obtained by letting n ! 1. The classical
theory of symmetric functions can therefore be regarded as the study of the polynomial
algebra K[ek : k (cid:21) 1] over an in(cid:12)nite sequence of indeterminates, graded by deg(ek) = k.
Note that the original variables xi can be eliminated from this de(cid:12)nition.
What makes this algebra interesting is the existence of many natural bases (labelled
by partitions), of a canonical scalar product, of a Hopf algebra structure, and of several
other algebraic operations, and also, its many interpretations in representation theory,
algebraic geometry, or mathematical physics.
First of all, Sym, as an algebra, has several distinguished sets of generators. Let
P
k
(cid:21)(t) = k(cid:21)0ekt be the generating series of elementary symmetric functions. Then,
the complete homogeneous functions hn can be de(cid:12)ned by their generating series (cid:27)(t) =
P P
(cid:0)1 k k(cid:0)1 d
(cid:21)((cid:0)t) = k(cid:21)0hkt , and the power-sums pn by (t) = k(cid:21)1pkt = dt log(cid:27)(t) (the
Newton formulas). One has as well Sym = K[h1;h2;:::] = K[p1;p2;:::].
A Hopf algebra structure is de(cid:12)ned by means of the comultiplication (cid:1)(pk) = pk (cid:10)
1+1(cid:10)pk, and the antipode !~(pk) = (cid:0)pk. It can be shown that Sym is isomorphic to its
graded dual.
The dimension of the homogeneous component of degree n of Sym is dimSymn =
jPart(n)j = p(n), the number of partitions of n. Linear bases of Sym are therefore
naturally labelled by partitions. The simplest ones are m(cid:21) (monomial symmetric func-
tions), e(cid:21);h(cid:21);p(cid:21) (products e(cid:21)1e(cid:21)2 (cid:1)(cid:1)(cid:1)e(cid:21)r etc.), and the Schur functions s(cid:21), which are the
fundamental ones in representation theory.
There is a canonical scalar product, which can be de(cid:12)ned by either of the following
equivalent formulas:
(cid:3)
hs(cid:21);s(cid:22)i = hm(cid:21);h(cid:22)i = hp(cid:21);p(cid:22)i = (cid:14)(cid:21)(cid:22)
(cid:3)
(where p(cid:22) = p(cid:22)=z(cid:22)). This scalar product materializes the self duality of Sym, in that (cid:1)
is the adjoint of the multiplication map f (cid:10)g 7! fg. The equivalence of these de(cid:12)nitions,
as well as the self duality, is a consequence of the following property
Y X
(cid:0)1
hu(cid:21);v(cid:22)i = (cid:14)(cid:21)(cid:22) , (1(cid:0)xiyj) = u(cid:21)(X)v(cid:21)(Y)
i;j (cid:21)
2
and of the classical Cauchy identity for Schur functions
X Y
(cid:0)1
s(cid:21)(X)s(cid:21)(Y) = (1(cid:0)xiyj) :
(cid:21) i;j
There is a second comultiplication de(cid:12)ned by (cid:14)(pk) = pk (cid:10)pk. This comultiplication
is dual to a multiplication known as the internal product (cid:3), i.e. hf (cid:3)g;hi = hf (cid:10)g;(cid:14)(h)i.
The internal product corresponds to the (pointwise) product of central functions on the
(cid:21) (cid:21)
symmetric group, via the Frobenius characteristic map (cid:31) 7! ch((cid:31) ) = s(cid:21), where the
(cid:21)
(cid:31) are the irreducible characters. This is an isometric isomorphism R(Sn) ! Symn,
where R(Sn) is the vector space spanned by the irreducible characters. One has then
ch((cid:30) ) = ch((cid:30))(cid:3)ch( ).
The Frobenius character formula expresses the value (cid:31)((cid:27)) of a character (cid:31) on a per-
(cid:21)
mutation (cid:27) of cycle type (cid:22) as (cid:31) ((cid:22)) = hs(cid:21);p(cid:22)i.
The cycle index is the map Z : KSn ! Symn, mapping a permutation of cycle type (cid:22)
onto p(cid:22), induces a canonical linear isomorphism between the center of the group algebra
and symmetric functions of degree n. Then, the character formula can be rewritten as
(cid:31)((cid:27)) = hch((cid:31)); Z((cid:27))i:
Another important point is the induction formula, interpreting the ordinary multi-
plication of symmetric functions in terms of induced representations. If f = ch((cid:24)) and
g = ch((cid:17)) are the characteristics of two representations of Sm and Sn, then fg = ch((cid:31))
where (cid:31) is the character of Sm+n induced by the character (cid:24)(cid:2)(cid:17) of its subgroup Sm(cid:2)Sn.
Schur functions can also be interpreted as characters of GL(n;C). The polynomial
representations V(cid:21) of GL(n;C) are parametrized by partitions of length at most n, and
s(cid:21)(x1;:::;xn) = traceV(cid:21)(X) where X is the diagonal matrix X = diag(x1;:::;xn).
Schur functions correspond in a similar way to the irreducible representations of the
q-deformed structures Hn(q), Uq(gln) or Fq(GLn), for generic q. For example, in the
case of the Hecke algebra Hn(q), which is the algebra generated by elements T1;:::;Tn(cid:0)1
satisfying the braid relations plus the quadratic ones (Ti +1)(Ti (cid:0)q) = 0, the character
formula can be written as [101, 9, 91]
(cid:21) (cid:0)`((cid:22))
(cid:31)(cid:22)(q) = hs(cid:21)(X); (q (cid:0)1) h(cid:22)((q (cid:0)1)X)i; (1)
where the \(cid:21)-ring notation" h(cid:22)((q (cid:0) 1)X) means the following. In general, if X and Y
are two (multi-) sets of variables, identi(cid:12)ed with the formal sum of their elements, the
symmetric functions of X +Y, X (cid:0)Y and XY are de(cid:12)ned by setting
pk(X+Y) = pk(X)+pk(Y); pk(X(cid:0)Y) = pk(X)(cid:0)pk(Y) pk(XY) = pk(X)pk(Y); (2)
and then by expressing any symmetric function as a polynomial in the power sums.
Therefore, the symmetric functions of (q (cid:0)1)X = qX (cid:0)X are the images of those of X
k
under the ring homomorphism pk 7! (q (cid:0)1)pk.
TheHecke algebraHn(q)wasintroducedbyIwahoriin[48]. Theoriginalde(cid:12)nitionwas
as follows. Let G = GL(n;Fq) and B be the subgroup of G formed by upper triangular
matrices. Then, G acts on the vector space M = C G=B spanned by the left cosets of B
n
(which canbe identi(cid:12)edwith complete(cid:13)agsinFq). To decompose this representation into
irreducibles, one can use Schur's lemma and look at the centralizer of CG in End(M). It
turns out that this centralizer is isomorphic to Hn(q).
3
From the knowledge of the irreducible representations of Hn(q), one can in principle
obtain the characters of the irreducible representations of G occuring in the spectrum of
M. The characters of these representations, now called unipotent representations, were
in fact (cid:12)rst obtained by Steinberg [102], by a di(cid:11)erent method, quite similar to the one
used by Frobenius to determine the characters of symmetric groups.
The unipotent representations are parametrized by partitions (cid:21) of n. Their name
comes from the fact that these are exactly as many as the conjugacy classes of unipotent
elements in G, which are also parametrized by partitions (cid:22) specifying the sizes of their
Jordancells,sothataunipotentcharacterisdeterminedbyitsvaluesonunipotentclasses.
(cid:21)
Steinberg's result was that the value (cid:31)(cid:22) of the unipotent character (cid:21) on the unipo-
tent class (cid:22) is a polynomial in q, which has since been recognized as a Kostka-Foulkes
polynomial [41, 76]
(cid:31)(cid:21)(cid:22) = K~(cid:21)(cid:22)(q): (3)
The Kostka-Foulkes polynomials are the coe(cid:14)cients of the transition matrices between
the bases of Schur functions and of Hall-Littlewood functions. The Hall-Littlewood Q-
functions are de(cid:12)ned, for N variables x1;:::;xN, by the orbit sums
!
(1(cid:0)t)`((cid:22)) X (cid:22) (cid:1)N(t)
Q(cid:22)(x1;:::;xN;t) = (cid:27) x (4)
[m0]t! (cid:27)2Sn (cid:1)N(1)
Q
m
where m0 = N(cid:0)`((cid:22)), [m]t = (1(cid:0)t )=(1(cid:0)t) and(cid:1)N(t) = i<j(txi(cid:0)xj). TheP-functions
are just scaled versions of the Q's
1
P(cid:22) = `((cid:22)) Q(cid:22); (5)
(1(cid:0)t) [m1]t!(cid:1)(cid:1)(cid:1)[mn]t!
where mi is the multiplicity of i in (cid:22). Then,
X
s(cid:21)(X) = K(cid:21)(cid:22)(t)P(cid:22)(X;t); (6)
(cid:22)
and
X
0
Q(cid:22)(X;t) = Q(cid:22)(X=(1(cid:0)t);t) = K(cid:21)(cid:22)(t)s(cid:21)(X) (7)
(cid:21)
0
where Q is the adjoint basis of P for the standard scalar product of Sym, and (cid:12)nally
X
Q~0(cid:22)(X;t) = tn((cid:22))Q0(cid:22)(X;t) = K~(cid:21)(cid:22)(t)s(cid:21): (8)
(cid:21)
The Hall-Littlewood functions can be compactly expressed by means of an action of
(cid:0)1
the Hecke algebra HN(q) on the algebra of polynomials in x1;:::xN (here, q = t ) [16].
This action is de(cid:12)ned in terms of the isobaric divided di(cid:11)erence operators (cid:25)i
xif (cid:0)(cid:27)i(xif)
(cid:25)i(f) = ; (9)
xi (cid:0)xi+1
where (cid:27)i is the ring involution exchanging xi and xi+1, by
Ti = (q (cid:0)1)(cid:25)i +(cid:27)i: (10)
4
The Hall-Littlewoodfunctions arethen, up toa scalarfactor, the imagesof themonomials
by the full symmetrizer
X
(N)
S = T(cid:27); (11)
(cid:27)2Sn
that is,
N
S(N)(x(cid:22)) = q(2)(1(cid:0)q(cid:0)1)(cid:0)`((cid:22))[m0]1=q!Q(cid:22)(x1;:::;xN;q(cid:0)1): (12)
The symmetric functions Q~0(cid:22) describe the characters of certain graded versions of
k
the permutation representations of Sn, i.e. the coe(cid:14)cient of q is the characteristic of
the homogeneous component of degree k (these are the Springer representations, in the
n
cohomology rings of unipotent varieties, see [47, 78, 63]). The simplest case (cid:22) = (1 ) is
already of interest. It describes the usual graded version of the regular representation,
which is realized for example in the coinvariants, in the cohomology of the (cid:13)ag manifold,
or in the space of Sn-harmonic polynomials. There is a closed formula for this graded
character !
X
Q~0(1n)(X;q) = (q)nhn (13)
1(cid:0)q
n
which implies that the Kostka-Foulkes polynomials for (cid:22) = (1 ) are essentially the prin-
cipal specializations of Schur functions, i.e.
K~(cid:21)(1n)(q) = (q)ns(cid:21)(1;q;q2;(cid:1)(cid:1)(cid:1)): (14)
The combinatorial properties of symmetric functions, e.g., the Littlewood-Richardson
ruleformultiplyingSchur functions, orthecombinatorialinterpretationofKostka-Foulkes
polynomials as generating functions of sets of Young tableaux according to a certain
statistic(charge orcocharge [68], see also[63]), relyessentiallyon the Robinson-Schensted
correspondence, which can be used to de(cid:12)ne a multiplicative structure on the set of
Young tableaux (the plactic monoid of Lascoux and Schu(cid:127)tzenberger [69]). The plactic
(cid:3)
monoid over an ordered alphabet A is the quotient of the free monoid A by the relations
xzy = zxy (x (cid:20) y < z), yxz = yzx (x < y (cid:20) z). These relations can now be understood
in terms of crystals. If one considers the letters of A as the vertices of the crystal graph of
the vector representation V of Uq(gln), and words of lengthN asthe vertices of the crystal
(cid:10)N
graph of V , then, two words are equivalent under the plactic relations i(cid:11) they label
corresponding vertices of two isomorphic connected components of the graph. This allows
to de(cid:12)ne a similar monoid for other Lie algebras, including the classical ones [66, 72].
The essential facts are that plactic classes correspond to Young tableaux, and that the
plactic Schur functions S(cid:21), de(cid:12)ned as the sum of all tableaux of the same shape (cid:21), span
a commutative subalgebra, isomorphic to Sym(x1;:::;xn). The Littlewood-Richardson
rule, for example, follows straightforwardly from these facts.
To conclude, let us mention that the main objects of current interest in the theory of
symmetric functions are the Macdonald polynomials [78]. We will not touch this subject
here, since the proper noncommutative analogues of Macdonald polynomialshave not yet
been worked out in general. However, the other properties that have been alluded to in
this section will all (cid:12)nd some kind of noncommutative analogue.
5
2 The Hopf algebra of noncommutative symmetric
functions
2.1 Algebraic generators
Imitating the description of Sym as a polynomial algebra on independent indeterminates
ek, graded by deg(ek) = k, one de(cid:12)nes the algebra Sym of formal noncommutative sym-
metricfunctionsasthefreeassociativealgebraonanin(cid:12)nitesequence (cid:3)k ofnoncommuting
indeterminates (the noncommutative elementary functions), graded by deg((cid:3)k) = k. The
coe(cid:14)cients are taken in some (cid:12)eld K of characteristic 0, which is assumed to contain
the rational functions in all extra variables t;q;z;::: used for generating functions or as
deformation parameters.
As in the commutative case, one introduces the generating series
X
n
(cid:21)(t) = (cid:3)nt ; (15)
n(cid:21)0
where t is an indeterminate in the ground (cid:12)eld K. The complete homogeneous symmetric
functions Sn are then naturally de(cid:12)ned as the coe(cid:14)cients of the series
X
(cid:0)1 n
(cid:27)(t) = (cid:21)((cid:0)t) = Snt : (16)
n(cid:21)0
A concrete realization Sym(A) of this algebra can be given by taking an in(cid:12)nite
sequence A = fanjn (cid:21) 1g of noncommuting indeterminates of degree 1, by setting
X Y
n
(cid:21)(A;t) = (cid:3)n(A)t = (1+tai); (17)
n(cid:21)0 i(cid:21)1
so that (cid:3)n(A) gets identi(cid:12)ed with the sum of allstrictly decreasing words of length n, and
Sn(A) with the sum of all nondecreasing words of the same length, which are respectively
represented as column-shaped and row shaped Young tableaux.
The algebra homomorphism Sym(A) ! Sym(X) de(cid:12)ned by ai 7! xi, called the
commutative image F 7! F, maps (cid:3)n to en, so that Sym is actually a noncommtative
liftingof the algebra of symmetric functions. One can object however, that the (cid:3)n(A) are
not invariant under the symmetric group. At least, not for the usual action. But we shall
see later that they are indeed symmetric for a more subtle one.
The (cid:12)rst really interesting question is \what are the noncommutative power sums ?".
There are indeed several possibilities. If one starts from the classical expression
8 9
<X tk=
(cid:27)(t) = exp pk ; (18)
: k ;
k(cid:21)1
one can choose to de(cid:12)ne noncommutative power sums (cid:8)k by the same formula
8 9
<X tk=
(cid:27)(t) = exp (cid:8)k ; (19)
: k;
k(cid:21)1
but a noncommutative version of the Newton formulas
nhn = hn(cid:0)1p1 +hn(cid:0)2p2 +(cid:1)(cid:1)(cid:1)+pn (20)
6
which are derived by taking the logarithmic derivative of (18) lead to di(cid:11)erent noncom-
mutative power-sums (cid:9)k inductively de(cid:12)ned by
nSn = Sn(cid:0)1(cid:9)1 +Sn(cid:0)2(cid:9)2 +(cid:1)(cid:1)(cid:1)+(cid:9)n: (21)
P
k(cid:0)1
Introducing the generating function (t) = k(cid:21)1(cid:9)kt , one may regard (cid:27)(t) as the
unique solution of the di(cid:11)erential equation
d
(cid:27)(t) = (cid:27)(t) (t) (22)
dt
satisfying the initial condition (cid:27)(0) = 1. The generating function of the (cid:8)k, taken in the
form
X tk
(cid:8)(t) = (cid:8)k (23)
k
k(cid:21)1
is then the logarithm of this solution. From this, one realizes that the relation between
the two kinds of noncommutative power-sums is of a rather complicated nature. The
expression of (cid:8)(t) as a function of the (cid:9)k is known as the continuous Baker-Campblell-
Hausdor(cid:11) series [80, 5, 84]. It can be written as a Chen series (iterated integrals) in a
quite explicit form (to be discussed later), and it is usually interpreted as expressing the
logarithm of the \evolution operator" (cid:27)(t) in terms of the \Hamiltonian" (t) [5]. It is
known that the continuous BCH series is a Lie series, so that the (cid:8)k are elements of the
free Lie algebra L generated by the (cid:9)k, of which they form another system of generators.
In fact, any sequence (Fn) of generators of L, with deg(Fn) = n can be shown to provide
an admissible family of noncommutative power sums, in the sense that the commutative
image of Fn is a nonzero multiple of pn. Moreover, the Poincar(cid:19)e-Birkho(cid:11)-Witt theorem
shows that Sym can be identi(cid:12)ed with the universal enveloping algebra U(L) of L. As
such, Sym is endowed with a canonical comultiplication (cid:1), for which L is the space of
primitive elements (Friedrich's theorem, see [94]). In particular,
(cid:1)(cid:9)k = (cid:9)k (cid:10)1+1(cid:10)(cid:9)k (cid:1)(cid:8)k = (cid:8)k (cid:10)1+1(cid:10)(cid:8)k; (24)
and also
Xn Xn
(cid:1)(cid:3)n = (cid:3)k (cid:10)(cid:3)n(cid:0)k; (cid:1)Sn = Sk (cid:10)Sn(cid:0)k: (25)
k=0 k=0
That is, the comultiplication of Sym is mapped onto the usual one on Sym under the
commutative image homomorphism.
There is also an analogue of the canonical involution ! : en $ hn, de(cid:12)ned in the same
n
way by !((cid:3)n) = Sn, and it is easy to check that the signed version !~((cid:3)n) = ((cid:0)1) Sn is
an antipode for (cid:1).
2.2 Linear bases
As for ordinary symmetric functions, we (cid:12)rst de(cid:12)ne linear bases by taking monomials
in the various families of algebraic generators, such as e(cid:21) = e(cid:21)1e(cid:21)2(cid:1)(cid:1)(cid:1)e(cid:21)r. Here, our
generators do not commute, so that basis elements of the homogeneous component Symn
of degree n will be labelled by compositions of n, i.e., ordered sequences I = (i1;:::;ir) of
positive integers. For a sequence (Gn) of homogenous generators with deg(Gn) = n, we
I I I I I
set G = Gi1Gi2(cid:1)(cid:1)(cid:1)Gir. Therefore, we already have the four bases (cid:3) , S , (cid:8) and (cid:9) .
7
A composition I of n is conveniently pictured as a ribbon diagram, which is a rim-
hook shaped skew Young diagram whose succesive rows have lengths i1;i2;:::;ir (read
from top to bottom in the French convention). For example, the ribbon diagram of shape
I = (3;2;1;4) is
n(cid:0)1
The number of compositionsof n is equal to 2 . A useful way to realizethis is to encode
the ribbon diagram of a composition I of n by the subset Des(I) = fi1;i1 +i2;:::;i1 +
i2+(cid:1)(cid:1)(cid:1)+ir(cid:0)1g of f1;2;:::;n(cid:0)1g. The elements of Des(I) are called the descents of the
composition.
The next basis that should be de(cid:12)ned, to pursue the parallel with the commutative
theory, would be the analogue of monomial symmetric functions. However, we have no
way to achieve this at this point. This is because in the classical case, the monomialbasis
(m(cid:21)) is dual to the homogeneous one (h(cid:21)), of which we already have the noncommutative
I
analogue (S ). But since the comultiplication (cid:1) of Sym is obviously cocommutative,
Sym cannot be self-dual, and the analogues of the monomial functions will have to live
(cid:3)
in the dual Hopf algebra Sym , to be discussed in the next section.
On another hand, we can de(cid:12)ne the analogues of Schur functions. These are the
so-called ribbon Schur functions. Their original de(cid:12)nition was given in terms of quasi-
determinants, but one can as well de(cid:12)ne them as follows.
The set of allcompositionsof a given integer n is equipped with the reverse re(cid:12)nement
order,denoted(cid:22). Forinstance, thecompositionsJ of4such thatJ (cid:22) (1;2;1)are(1;2;1),
(3;1), (1;3) and (4). The ribbon Schur function (RI) is de(cid:12)ned by the alternating sum
0 1
X X
RI = ((cid:0)1)`(I)(cid:0)`(J)SJ @() SI = RJA ; (26)
J(cid:22)I J(cid:22)I
where `(I) denotes the length of I. Clearly, (RI) is also a homogenous basis of Sym.
The commutative image of a ribbon Schur function RI is the corresponding ordinary
ribbon Schur function, which will be denoted by by rI. The rI were (cid:12)rst introduced
by MacMahon, in his analysis of Simon Newcomb's problem [79]. They arise also, for
example, in a generalization by Lascoux and Pragacz [67] of the Giambelli formula (ex-
pressing general Schur functions as determinants of ribbons instead of just hooks), or
as sln-characters of the irreducible components of the Yangian representations in level 1
b
modules of sln [54].
AnimportantpropertyoftheribbonSchurfunctionsistheirverysimplemultiplication
formula (already known to MacMahon in the commutative case)
RI RJ = RI.J +RI(cid:1)J (27)
where I.J denotes thecomposition(i1;:::;ir(cid:0)1;ir+j1;j2;:::;js)andI(cid:1)J thecomposition
(i1;:::;ir;j1;:::;js).
The transition matrices between the above bases can be worked out quite explicitely
(see [32]).
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