Lectures on Noncommutative Symmetric Functions

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 parallel with the commutative theory and on the representation theoretical interpretations. Some examples involving descent algebras and characters of symmetric groups are discussed in detail. Introduction The theory of noncommutative symmetric functions is the outgrowth of a program initiated 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 analogue of 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 enveloping algebras, roots of noncommutative polynomials, noncommutative continued fractions, Pad e aprroximants or orthogonal polynomials [32, 85, 35, 36]. These calculations usually take place in a skew eld, which is the 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 representation 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 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 rst one, involving Hecke algebras at q = 0, leads us to the de 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 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 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. These notes correspond to a series of lectures delivered at the workshop \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 commutative symmetric functions. A minor change is that the algebra of symmetric functions is denoted by Sym, the coe cients being taken in some 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 Sym(X) = K [x1 ; : : : ; xn]Sn (K is some eld of characteristic 0) is freely generated by the elementary symmetric polynomials ek = X i1<::: w(i+ 1), whence the terminology. In this case, we rather say that i is a descent of w. Let Des (w) denote the descent set of w, and for a subset E S, set DE = X Des (w)=Ew 2 ZW : (48) 11 Then, Solomon's result is that the DE span a Z-subalgebra of ZW . Moreover DE0DE00 =XE cEE0E00DE (49) where the coe cients cEE0E00 are nonnegative integers. The canonical anti-isomorphism : n ! Symn maps the descent class DE to the ribbon Schur function RI , with I such that E = Des (I). From one of Solomon's formulas, one obtains the following multiplication rule. Let i = (i1; : : : ; ip) and J = (j1; : : : ; jq) be two compositions of n. Then, SI SJ = X M2Mat (I;J) SM (50) where Mat (I; J) denotes the set of matrices of nonnegative integers M = (mij) of order p q such that Ps mrs = ir and Pr mrs = js for r 2 [1; p] and s 2 [1; q], and where SM = Sm11 Sm12 Sm1p Smq1 Smqp : Note that by de nition, if F and G are homogeneous of di erent degrees, F G = 0, and that Sn is the unit element of the -subalgebra Symn. Let h = hI = SI and h = hJ = SJ be the commutative images of SI and SJ . From the known expression of h h in the commutative case, one can see that (SI SJ) = SI SJ , so that in general F G = F G ; (51) that is, the commutative image is a homomorphism for the internal products. From equation (50), one derives a fundamental formula, whose commutative version is just a special case of the Mackey formula for the restriction of an induced character. Let F1; F2; : : : ; Fr; G 2 Sym. Then, (F1F2 Fr) G = r [(F1 Fr) rG] (52) where in the right-hand side, r denotes the r-fold ordinary multiplication and stands for the operation induced on Sym n by [32]. In the commutative case, the power-sums pn and more generally the power-sum products are quasi-idempotents (i.e., idempotents up to a scalar factor) for the internal product. Precisely, p p = z p (53) where for = (1m12m2 nmn), z = Qi imi mi!. Therefore, the commutative images of noncommutative power sums and their products are quasi-idempotents, and one may wonder whether there are true quasi-idempotents among them. Thanks to the antiisomorphism with the descent algebra, we could then use them to construct idempotents in the group algebras of symmetric groups. As an illustration of the formalism, let us try this program with the power sums n. We want to know whether n n = n n. To this end, we can write a generating function for the -squares in the form X n 1(xy)n 1( n n) = (x) (y) ; (54) 12 since i j = 0 for i 6= j. Now, writing (22) in the form (t) = X n 1 tn 1 n = ( t) 0(t) and applying the splitting formula (52), we get X n 1(xy)n 1( n n) = ( x) 0(x) ( y) 0( y) = [( ( x) 0(x)) ( (y) 1 + 1 (y))] = [( ( x) 1) ( 0(x) (y))] ; (since 0(x) has no constant term) = 0@ X n 1 nxn 1 Sn1A 0@ X n 1 yn 1 n1A = X n 1 (xy)n 1 n n ; the last equality following from the fact that Sn F = F for F 2 Symn. Hence, n n = n n, so that n = 1( n) is a quasi-idempotent of n. To see what it looks like, we have to expand n on the ribbon basis. The linear recurrence (21) together with the multiplication formula for ribbons (27) (recall that Sk = Rk) easily yields n = Rn R1;n 1 +R1;1;n 2 = n 1 X k=0( 1)kR1k;n k ; (55) which is analogous to the classical expression of pn as the alternating sum of hook Schur functions. Therefore, in the descent algebra, n = n 1 X k=0 ( 1)kDf1;2;:::;kg : (56) On this expression, we can recognize a famous element of the group algebra of the symmetric group, namely, Dynkin's left-bracketing operator [20] (see also [100, 109, 29, 2, 94]). The standard left bracketing of a word w = x1x2 xn is the Lie polynomial Ln(w) = [ [[[x1; x2]; x3]; x4]; : : : ; xn] : (57) This formula de nes a linear operator Ln on the homogeneous component K nhAi of the free associative algebra K hAi. In terms of the right action of the symmetric group Sn on K nhAi, de ned on words by x1 x2 xn = x (1) x (2) x (n) ; (58) one can write Ln(w) = X 2Sn a (w ) = w n the coe cient a being 1 or 0, according to whether is a \hook permutation" or not. To see this, one just has to write the permutations appearing in the rst i as ribbon tableaux and then to argue by induction. For example, 3 = [[1; 2]; 3] = 123 21 3 31 2 + 321 13 and it is clear that when expanding 4 = [ 3; 4] one will only get those (signed) tableaux obtained from the previous ones by adding 4 at the end of the last row, minus those obtained by adding 4 on the top of the rst column. Thus, we have proved that 1 nLn is a projector, whose image is obviously a subspace of the free Lie algebra. By iteration of Jacobi's identity, it is easy to see that any Lie element can be written as a linear comnbination of standard left bracketings, so that what we have actually obtained is a proof of Dynkin's characterization of Lie elements: a noncommutative polynomial P 2 K hAi is a Lie polynomial i Ln(P ) = nP [20, 94, 7]. Idempotents such as 1 n n, acting as projectors onto the free Lie algebra are usually called Lie idempotents [29, 2, 94]. They play an important role in the analysis of the Hausdor series, or as the building blocks of other idempotents, such as the Eulerian idempotents, used in Hodge-type decompositions of certain cohomology theories (see [38, 73, 75, 90, 88, 89]). So far, our formalism has just led us to an exotic proof of a classical result. Let us now see whether the method contains the germ of some generalization. One ingredient in our proof was the analogue (55) of the expansion of a power sum as an alternating sum of hook Schur functions. This expression has a well known q-analogue, namely, the one involved in the character formula for Hecke algebras (1). It can be written in the form hn((1 q)X) 1 q = n 1 X k=0( q)ksn k;1k : (59) Let us look for a noncommutative analogue of this expression. To this aim, it will be convenient to extend as much as possible the -ring notation of the classical theory. Given two totally ordered sets A and B of noncommuting variables, we can de ne the virtual alphabet A B by specifying its complete symmetric functions Sn(A B). Their generating series is de ned by (A B; t) := ( B; t) (A; t) : (60) One also de nes the symmetric functions of A+B by (A +B; t) := (A; t) (B; t) (61) where now, A and B can be either real or virtual, and for a real ordered commutative alphabet X, the virtual alphabet XA is de ned by (XA; t) = ! Y i 1 (A; txi) : (62) These de nitions imply a de nition of quasi-symmetric functions of a di erence f(X Y ), which is the same as the one obtained by composing the comultiplication and the antipode, and we can now give a meaning to the noncommutative symmetric functions of all virtual alphabets of the type (X Y )(A B). The case we have in mind corresponds to X = f1g and Y = fqg. According to our de nitions, ((1 q)A; t) = (A qA; t) = (qA; t) (A; t) = Pk 0 tk( q)k k(A)Pl 0 tlSl(A) : 14 Taking into account the fact that k = R1k and applying the multiplication rule for ribbons, we obtain Sn((1 q)A) = (1 q) n 1 X k=0( q)kR1k ;n k ; (63) the q-analogue we were looking for. The symmetric function n(q) := Sn((1 q)A) 1 q (64) corresponds to a natural q-analogue of Dynkin's element in the descent algebra. Indeed, writing permutations as words in the letters 1; 2; : : : ; n, it is easy to see that the image of n(q) under the isomorphism 1 is the left q-bracketing of the standard word 1 2 : : : n, that is, 1( n(q)) = n(q) = [[: : : [[ 1; 2 ]q; 3 ]q; : : : ]q; n ]q where [R; S]q = RS q SR. Now, we can prove the following q-analog of Dynkin's theorem, which, according to what we have already seen, can be understood as describing the left q-bracketing of a homogeneous Lie polynomial: n(q) n = [n]q n : (65) The proof works exactly as the previous one. Let #(t) = X n 1 n(q)tn 1 = ((1 q)A; t) 1 (1 q)t : (66) It is easy to see that #(t) = (A; qt) 0 q(A; t) (67) where 0 q denotes the the q-derivative 0 q(t) := (t) (qt) (1 q)t (68) with respect to t. Then, one can write #(t) n = (( (A; qt) 0 q(A; t)) ( n)) = (( (A; qt) 0 q(A; t)) (1 n+ n 1)) ; from which it follows that #(t) n = ( (A; qt) 1) ( 0 q(A; t) n) = [n]q n tn 1 : Equation (65) means that homogeneous Lie polynomials of degree n are again eigenvectors of the left q-bracketing operator, now with the q-integer [n]q as eigenvalue. They actually constitute the [n]q eigenspace. However, the q-Dynkin element n(q) is invertible in the group algebra for generic values of q, and its other eigenvalues are nonzero. It would therefore be of interest to investigate its spectral decomposition. This will be done in Section 4 In the last two examples, we have been interpreting noncommutative symmetric functions as linear operators on the free algebra K hAi, by means of the identi cation of Symn 15 with n and of the right action of permutations. If we extend the action of Sn to all words by deciding that w = 0 if w is not of length n, we obtain in this way a representation : Sym ! Endgr(K hAi) (69) where Endgr is the algebra of degree-preserving endomorphism. Under the representation , the internal product is mapped to the composition , and, as shown by Reutenauer [94], the ordinary product becomes the convolution product ? of Endgr(K hAi), which is de ned by f ? g = m (f g) C (70) where m : u v 7! uv is the multiplication map of K hAi and C is its standard comultiplication, de ned by C(x) = x 1 + 1 x for x 2 A and C(uv) = C(u)C(v).