Weights of Markov Traces on Hecke Algebras. R.C. Orellana y. November 6, Abstract

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1 Weights of Markov Traces on ecke Algebras R.C. Orellana y November 6, 998 Abstract We compute the weights, i.e. the values at the minimal idempotents, for the Markov trace on the ecke algebra of type B n and type D n. In order to prove the weight formula, we dene representations of the ecke algebra of type B onto a reduced ecke Algebra of type A. To compute the weights for type D we use the inclusion of the ecke algebra of type D into the ecke algebra of type B. Introduction The ecke algebra of type B n, n (q; Q), is semisimple whenever Q 6=?q k, k 2 f0; ; : : : (n? )g, and q is not a root of unity, see [DJ]. The simple components are indexed by ordered pairs of Young diagrams. These ecke algebras can be dened as a nite dimensional quotient of the group algebra of the braid group of type B. Motivated by their study of link invariants related to the braid group of type B, Geck and Lambropolou [GL] have dened certain linear traces on the ecke algebra of type B called Markov traces. Their denition is given inductively. In this paper we give an alternative way of computing this trace. Since the ecke algebra of type B is semisimple, any linear trace can be written as a weighted linear combination of the irreducible characters (the usual trace). The coecients in this linear expression are called weights. The weights are equal to the values of the trace at the minimal idempotents. Since the characters are known, it follows that the weights completely determine the trace. The weights are also indexed by ordered pairs of Young diagrams. We have found the weight formula for the Markov trace dened by Geck and Lambropolou [GL] for the ecke algebra of type B. The weight formula can be written as a product of Schur functions. To prove this formula we construct a homomorphism from the specialization of the ecke algebra of type B, n (q;?q r +m ), onto a reduced ecke algebra of type A. Using 99 Subject Classication: 0599 y Partially Supported by President's Dissertation Fellowship and NSF grant DMS

2 2 R. C. Orellana this homomorphism we obtain that the Markov trace of the ecke algebra of type B appears as a pullback of the Markov trace of the reduced ecke algebra of type A. A consequence of the above mentioned homomorphism is the existence of a duality between the quantum group U q (sl(r)) and the specialized ecke algebra n (q;?q r +m ). In order to nd the weights for the ecke algebra of type D we use the results of oefsmit [] on the inclusion of the ecke algebra of type D into the ecke algebra of type B. We also use the results of Geck [G] on obtaining Markov traces of the ecke algebra of type D from those of type B. In a future paper we will use the weight formula and the onto homomorphism to determine the structure of a certain quotient of the ecke algebra at roots of unity. This quotient of the ecke algebras is important in the theory of von Neumann algebras since they provide examples of subfactors of the hypernite II factor. This paper is organized as follows. In the rst section we give some background denitions for partitions and Schur functions. We also dene the braid groups of type A and type B; and we give representations of the braid group of type B into the braid group of type A. In the second section we dene the ecke Algebra of type B. We also dene the right coset representatives and distinguished double coset representatives of n? (q; Q) in n (q; Q) which are needed in the later sections. In the third section we show that the representations given in section for braid groups can be extended to well-dened representations of the ecke algebra of type B onto a reduced ecke algebra of type A. In the fourth section we dene the Markov trace and quote some of the results from [GL] which are relevant in our quest. In the fth section we prove the weight formula. In the last section we derive the weights for the ecke algebra of type D from the weights for the ecke algebra of type B. Acknowledgments: This paper is a part of my Ph.D. thesis at University of California, San Diego. I would like to thank my advisor ans Wenzl for all the encouragement, for all the useful suggestions and for the nancial support. Preliminaries and Notation Let F be a eld of characteristic 0. The algebra of n n matrices over F is denoted by M n (F L L m ) = M n. Let A B be an inclusion of semisimple algebras. Then A = A i= i and l B = B j= j, with A i = Mai, B j = Mbj and a i ; b j 2 N. Since a simple B j module is also an A module, it can be decomposed into a direct sum of simple A modules. Let g ij be the number of simple A i modules in this decomposition. The matrix G = (g ij ) is called the inclusion matrix for A B. This matrix can be conveniently described by an inclusion diagram. This is a graph with vertices arranged in 2 lines. In one line, the vertices are in? correspondence with the simple summands A i of A, in the other

3 Weights of Markov Traces on ecke Algebras 3 one with summands B j of B. Then a vertex corresponding to A i is joined with a vertex corresponding to B j by g ij edges. This is known as the Bratteli diagram. A trace is a linear funtional tr : B! F such that tr(ab) = tr(ba) for all a; b 2 B. Recall that tr is nondegenerate if for any b 2 B, b 6= 0, there is some b 0 2 B such that tr(bb 0 ) L 6= 0. There is up to scalar multiples only one trace on M n (F ). Any trace tr on B = B j is completely determined by a vector ~t = (t j ), where t j = tr(p j ) and p j is a minimal idempotent of B j. The vector ~t is called the weight vector of tr. In a similar manner we can characterize a trace on A by a weight vector ~s = (s i ). The weight vectors ~t and ~s are related as follows: G~t = ~s:. Partitions and Standard Tableaux A partition of n is a sequence of positive numbers = ( ; : : : ; k ) such that 2 : : : k 0 and n = jj = : : : + k. The i 's are called the parts of, and the number of nonzero parts of is called the length of, denoted by l(). If we say l() r, then we we will mean that there are s r nonzero parts and the remaining r? s are equal to zero. We use the notation ` n to mean is a partition of n. A Young diagram is a pictorial representation of a partition as an array of n boxes with boxes in the rst row, 2 boxes in the second row, and so on. We count the rows from top to bottom. We shall denote the Young diagram and the partitions by the same symbol. We use the word shapes interchangeably with the word partitions. The set of all partitions of n is denoted by n. If and are two partitions with jj jj, then we write if i i for all i. In this case we say that is contained in. If then the set-theoretic dierence? is called a skew diagram, =. A standard tableaux of shape is a lling of the boxes with numbers ; 2; : : : ; n such that the numbers in each row increase from left to right and in every column from top to bottom. The notation t will be used to denote a standard tableaux of shape. We say t is contained in t, denoted t t, if t is obtained by removing appropriate boxes from t, i.e. the numbers ; 2; : : : ; jj are in the same boxes of both in t and in t. A double partition of size n, (; ), is an ordered pair of partitions and such that jj + jj = n. The length of a double partition is l(; ) = l() + l(). A double partition can be associated with a pair of Young diagrams. The set of all pairs of Young diagrams with n boxes is denoted by D n. We dene as for single partitions (; ) (; ) if i i and j j for all i and j. That is, (; ) is obtained from (; ) by removing appropriate boxes of (; ), and we say (; ) is contained in (; ). t (;) = (t ; t ) is a pair of standard tableaux if the arrangement of the numbers is in increasing order in the rows and columns of both t and t. For any double partition (; ),

4 4 R. C. Orellana let T (;) denote the set of standard tableaux of shape (; ) and T n denote the set of all pairs of tableaux with n boxes. For example, the diagram below is a pair of standard tableaux of shape ([2; ; ]; [3; 2]) (, 5 8 ) 4 Figure We say that a box in (; ) has coordinates (i; j) if the box is in the i-th row and j-th column of either or. Two boxes in (; ) can have the same coordinates if they occur in the same box in as in ; for instance, the left-top-most box in and the left-top-most box in both have coordinates (; ). The following observation will be used throughout the sequel. For m; r ; n 2 N, let m > n and r > n. Consider a pair of Young diagrams (; ) with n boxes. We construct from the pair (; ) a Young diagram with mr + n boxes by adjoining a rectangular box with r rows and m columns, as in Figure 2. The diagram described corresponds to the following partition: = [m + ; m + 2 ; : : : ; m + r ; ; 2 ; : : : ; r2 ] Observation: Let n, m, and r be as above. Let [m r ] ` f. Then there is a - correspondence between pairs of Young diagrams with n boxes and Young diagrams containing [m r ] with n + f boxes. m - 6 r!? Figure 2

5 Weights of Markov Traces on ecke Algebras 5.2 Schur Functions For details and proofs on the results mentioned in this subsection we refer the reader to [M] Ch.I, sec. 3. Suppose x ; x 2 ; : : : x r is a nite set of variables. For some partition we set x = x : : : x r. Let = [r? ; r? 2; : : : ; ; 0]. We dene the following determinant. r a + = det(x j+r?j i ) This determinant is divisible in Z[x ; : : : ; x r ] by the Vandermonde determinant which is given by a = det(x r?j i ): Then the symmetric Schur function is dened as follows: s (x ; : : : ; x r ) = a + a : Take x i = q i? ( i r). If is a partition of length r, we have s (; q; : : : ; q r? ) = q n() Y i<jr? q i? j +j?i? q j?i () where n() = P l() i= (i? ) i. We dene the following Schur function as a normalization of equation (): s ;r (q) = s (; q; : : : ; q r? ) s [] (; q; : : : ; q r? ) jj = q n() Y i<jr (? q)(? q i? j +j?i ) (? q r )(? q j?i ) Notice that s ;r (q) = 0 whenever l() > r. Also if (q) denotes the character of a matrix in Gl(r) with eigenvalues q (?r+)=2, q (?r+3)=2, : : :, q (r?)=2 and if the number of boxes in is n, s ;r (q) = (q)=( [] (q)) n. Let r = r + r 2 ; then the Schur function for = [m + ; : : : ; m + r ; ; : : : ; r2 ] is given by s ;r (q) = q n() Y i<jr (? q)(? q i? j +j?i ) : (2) (? q r )(? q j?i ) After some rearrangement of equation (2) we get the following equation for the Schur function of : s ;r (q) = q mr (r?)=2+r jj (? q? q r )jj s (; q; : : : ; q r? )s (; q; : : : ; q r 2? ) r Y r 2 Y i= j=? q m+r + i? j +j?i : (3)? q r +j?i The expression of this Schur function in (3) is useful in the proof of the weight formula!

6 6 R. C. Orellana.3 The Braid Group The braid group of type A, B n (A), can be dened algebraically by generators ~ ; ~ 2 ; : : : ; ~ n? and relations ~ i ~ j = ~ j ~ i if ji? jj > ; (4) ~ i ~ i+ ~ i = ~ i+ ~ i ~ i+ if i n? 2: (5) Similarly we can dene the braid group of type B, B n (B), by generators t; ; 2 ; : : : ; n? and dening relations given by equations (4) and (5) and t t = t t ; (6) t i = i t if i > : (7) In Figure 3 we illustrate the generator ~ i 2 B n (A), the full-twist B 3 (A), and the generators t; i 2 B n (B). i ~ i i+ n D 2 D 3 D ((( D D D ((( D D D D D D D ((( D D D D D D D 2 3 Figure 3 2 t n i i+ In general, the full-twist 2 in f f strings, is a central element in B f(a). Algebraically, 2 = f (~ f? : : : ~ ) f. We now dene a map from the generators of B n (B) into B f +n (A). We call the map ~ f;n, and we dene the image of the generators of B n (B) as follows: ~ f;n (t) = 2 and f ~ + f;n( i ) = ~ f +i for i = ; : : : ; n. Pictorially, we have the following:?2 f i n 2 n 2 f f+ f+2 f+n i i+ n f+i f+i+ f+n t ~ f;n -?2 f 2 f + ~ f;n - i ~ f +i Figure 4

7 Weights of Markov Traces on ecke Algebras 7 Proposition. Let n; f 2 N and ~ f;n be as dened above for the generators of B n (B). Then ~ f;n is a well-dened group homomorphism. Proof: It suces to prove that ~ f;n preserves the relations of B n (B). It is convenient to express ~ f;n (t) = ~ f : : : ~ 2 ~ 2 ~ 2 : : : ~ f. Relations (4),(5) and (7) follow immediately from the denition of ~ f;n and the denition of B n (A). To show that (6) holds we will use the commuting property of the full-twist. f + ( 2 f +?2 f ) f +( 2 f +?2 f ) = 2 f +2?2 f + 2 f +?2 f = 2 f +?2 f f + 2 f +?2 f f +: It might be easier to see that this relation holds from the picture, see Figure 5. This completes the proof. f f+ f+2 f+3 f+n f f+ f+2 f+3 f+n = ~ f + (?2 f 2 f + )~ f +(?2 f 2 f + ) (?2 f 2 f + )~ f +(?2 f 2 f + )~ f + Figure 5 Corollary.2 The representations f;n can be extended in a natural way to representations of the corresponding braid group algebras. 2 ecke Algebra of type B n In this context, we will mean by the ecke algebra n (q; Q) of type B n the free complex algebra with and generators t; g ; g 2 ; : : : ; g n? and parameters q; Q 2 C with dening relations: () g i g i+ g i = g i+ g i g i+ for i = ; 2; : : : ; n? 2; (2) g i g j = g j g i, whenever j i? j j 2; (3) g 2 i = (q? )g i + q for i = ; 2; : : : ; n? ; (4) t 2 = (Q? )t + Q; (5) tg tg = g tg t;

8 8 R. C. Orellana (6) tg i = g i t for i 2. Note that for q = and Q =, (3) and (4) become g 2 i = and t 2 =, respectively. In this case the generators satisfy exactly the same relations as a set of simple reections of the hyperoctahedral group, n. It is known that n (q; Q) = C n (the group algebra of n ) if q is not a root of unity and Q 6=?q s for?n < s < n. C n is semisimple since it is a nite complex group algebra. This implies that n (q; Q) is semisimple for generic values of q and Q. Furthermore the simple modules are in - correspondence with double partitions. Their decomposition rule and their dimensions are the same as for n. [See Bourbarki, Groups et algebres de Lie IV, V, VI] oefsmit, in [], has written down explicit irreducible representations of n (q; Q) for each ordered pair of Young diagrams. This demonstrates that the dimension over C (q; Q) of n (q; Q) is 2 n n!. Let V (;) be an n (q; Q)-module then we have the following restriction rule: V (;) = n? (q;q) M (;) 0 (;) V (;) 0; (8) where (; ) 0 is the double partition obtained from (; ) by subtracting one from one Sof the parts. This isomorphism follows from the bijection t (;)! t (;)0 of T (;) and T (;) 0 (;) (;) 0, where t(;)0 is the pair of standard tableaux obtained by removing the box containing n. Moreover, (n) = L (;)`n (;) is a faithful representation of n (q; Q). Note that () and (2) are the dening relations for the braid group B n (A). So representations of n (q; Q) also yield in particular representations of B n (A). Obviously we also obtain representations of the braid group of type B. The ecke algebras satises the following embedding of algebras 0 2 We shall denote (q; Q) := [ n0 n (q; Q) Observe that (3) and (4) imply that t and g i have at most 2 eigenvalues each, hence also at most 2 projections corresponding to these eigenvalues. Right and Double Coset Representatives The fact that q is invertible in C (q; Q) implies that the generators g i are also invertible in n (q; Q). In fact, the inverse of the generators is given by g? i = q? g i + (q?? ) 2 n (q; Q) This implies that the following element is well-dened in n (q; Q): t 0 i = g i g tg? g? i. We use this elements to dene the set D n as a subset of n (q; Q). If n =, we let D := f; tg. For n 2 we have D n := f; g n? ; t 0 n?g:

9 Weights of Markov Traces on ecke Algebras 9 These are known as the distinguished double coset representatives of n? (q; Q) in n (q; Q). Also the set of right coset representatives of n? (q; Q) in n (q; Q) is dened as follows: R n := f; t 0 n? ; g n?g n?2 : : : g n?k ; g n? g n?2 : : : g n?k t 0 n?k? ( k n? )g: Note that R = f; tg. The elements fr r 2 : : : r n jr i 2 R i g form a C (q; Q)-basis of n (q; Q), see [GL]. 3 Representations of the ecke algebra of type B onto a reduced ecke algebra of type A In this section we will show that for a specialization of the ecke algebra of type B, we have a homomorphism onto a reduced ecke algebra of type A. But before we prove this assertion we introduce some necessary background. In this section we assume that for a xed integer n and m; r 2 N we have m > n and r > n. For these integers we choose = [m r ]. Then as we observed in Section, there is a one-to-one correspondence between double partitions of n and partitions of n + mr containing. Once we choose we x a standard tableau t. We also assume throughout that q is not a root of unity. The representations of n (q; Q) dened by oefsmit, (see []), depend on rational functions with denominators (Qq d + ) where d 2 f0; ; : : : ; (n? )g. So if Q =?q r +m then? q r +m+d 6= 0 as long as d 6=?(r + m), which implies that all representations of n (q;?q r +m ) are well-dened as long as r + m > n. Thus the specialized algebra, n (q;?q r +m ) is well-dened and semisimple. 3. The ecke algebra of type A By the ecke algebra of type A n?, n (q), we mean the free complex algebra with generators ~g, : : :, ~g n? and with parameter q 2 C and dening relations (f ) ~gi ~g i+ ~g i = ~g i+ ~g i ~g i+ for i = ; 2; : : : ; n? 2; (f 2) ~gi ~g j = ~g j ~g i, whenever j i? j j 2; (f 3) ~g 2 i = (q? )~g i + q for i = ; 2; : : : ; n? ; n (q) is semisimple whenever q is not a root of unity. If ` n, then ( ; V ) denotes the irreducible representation of n (q) indexed by. ere V is the vector space with orthonormal basis given by fv t g where t is a standard tableau of shape. These representations can be considered as q-analogs of Young's orthogonal representations of the Symmetric group (see [] or [W]). We now describe these representations in more detail.

10 0 R. C. Orellana Let d(t ; i) = c(i + )? c(i)? r(i + ) + r(i) where c(i) and r(i) denote respectively, the column and row of the box containing the number i, and for d 2 Zn f0g let a d (q) = qd (? q)? q d and c d (q) = Then one denes on the vector space V by (~g i )v t = a d (q)v t + c d (q)v ~gi (t ) p (? q d+ )(? q d? )? q d : where d = d(t ; i) and ~g i (t ) is the tableau obtained by interchanging i and i + in t. This representation follows from the ones dened in [W] by setting (~g i ) = (q + ) (e i )? Id V, where e i is the eigenprojection corresponding to the characteristic value q of ~g i. Moreover n = L `n is a faithful representation of n (q). Thus we have that n (q) = L `n ( n (q)), where ( n (q)) = M f and M f is the ring of complex f f matrices, and f is the number of standard tableaux of shape. If is a partition of n then denote by 0 the partition obtained by decreasing one part of by one; and 00 is obtained by decreasing one part of 0 by one. Let t be a standard tableau of shape, then t! (t ) 0 denes a bijection between T and S 0 T 0, where (t ) 0 is the Young tableau obtained by removing the box containing n from t. This in turn yields the following isomorphism of modules V M = V 0: (9) n? (q) 0 A special element in f (q) is the full-twist dened algebraically by 2 f := (~g f? : : : ~g ) f. The full-twist is an element in the center of f (q). The action of 2 f on ( ; V ) is described in the following lemma, the proof of which appears in [W2], pg. 26. Lemma 3. Let ` f. Then the full-twist acts by a scalar on the irreducible representation ( ; V ) of f (q) where = q f (f?)?p i<j ( i+) j : 3.2 A omomorphism of n (q;?q r +m ) onto a reduced algebra of n+f (q) In ([W], Cor. 2.3) Wenzl dened a special set of minimal idempotents of f (q) indexed by the standard tableaux. The sum of these idempotents is. These minimal idempotents are well-dened whenever f (q) is semisimple. If p 2 f (q) is an idempotent then the reduced algebra of f (q) with respect to this idempotent is p f (q)p := fpap j a 2 f (q)g. Let ` f and t be a standard tableau of

11 Weights of Markov Traces on ecke Algebras shape. Then p t is the minimal idempotent indexed by t. Thus, the reduced algebra decomposes as follows: p t n+f (q)p t = M ; `n+f (p t ) ( n+f (q)) (p t ): Notice that if is a partition of n + f and does not contain then (p t ) is the zero matrix. In particular, if we choose to be rectangular, we have that the reduced algebra p t f + (q)p t has only two nonzero irreducible modules indexed by partitions [m + ; m r? ] and [m r ; ], since these are the only partitions of f + which contain. Recall that in Section we showed that there is a homomorphism ~ f;n from the braid group B n (B) into the braid group B f +n (A). In what follows we will show that it is possible to extend ~ f;n to a homomorphism of the corresponding ecke algebras. Lemma 3.2 For a xed integer n and m; r 2 N, let m > n and r > n. Set f = mr and assume = [m r ] and = [mr ; ]. Let and be as in Lemma 3.. Choose a minimal idempotent p t in f (q). We dene a map f;n for the generators of n (q;?q r +m ) as follows: f;n () = p t ; f;n (t) =? p t?2 f 2 f + and f;n (g i ) = p t ~g i+f for i = ; : : : ; n?, (see Figure 6 for a pictorial denition). Then f;n extends to a welldened homomorphism of algebras, f;n : n (q;?q r +m )! p t n+f (q)p t. Proof: It is enough to check that f;n preserves the relations of the ecke algebra of type B. First notice that since p t 2 f (q), it commutes with all ~g j for all j > f. From this observation and the dening relations of n+f (q) it follows that we only need to check the relations involving f;n (t). For relations (5) and (6), (see pg. 7), notice that p t commutes with the full-twist 2, f +?2 f and with ~g f +. Thus these two relations will hold by the proof of Prop..2. It remains to be shown that f;n (t) has two eigenvalues:? and Q =?q r+m. In particular, we want to show that f;n (t) acts by a scalar on two irreducible modules of f + (q) and by 0 on all others. Since is rectangular this means that there are only two partitions of f + containing it. By lemma 3. we have that the full-twist, 2, acts by the scalar f + (resp. ) on the irreducible module indexed by (resp. ). Also?2 f acts by? on the irreducible modules of f + (q) which are indexed by Young diagrams which contain. Therefore, we have that?2 f 2 acts by f +? on the module V and by? on the module V. Therefore, we have that? p t?2 f 2 acts by? on f V + and by? on V, and by zero on all other modules, since p t kills all modules which do not contain.

12 2 R. C. Orellana In order to determine the constant? we only need to substitute the partitions = [m + ; m r? ] and = [m r ; ] in the formula given in lemma 3.. Thus we obtain? =?q?p i<j ( i+) j + P i<j ( i+) j =?q r +m : This concludes this proof. The following gure gives a pictorial denition of the homomorphism f;n dened in the previous lemma. q 2 n q q f;n - p t p t f f+ f+2 f+n i Figure 6 i+ n f;n - p t p t f f+ f+i f+i+ f+n We have shown that f;n is a homomorphism. In the remainder of this section we will show that it is onto. In particular, we will show that the irreducible representations of the reduced algebra are also irreducible representations of n (q;?q r +m ). Notice that for ` n + f there is a - correspondence between standard skew tableaux of shape = and tableaux t which contain t. For this reason we denote by T = the set of standard tableaux which contain t. Notice that the order of T = is equal to the number of standard skew tableaux of shape =. For our choice of m, i.e. m > n, we have that = will consist of two parts which? can be interpreted as a double partition, say (; ). In this n case, the order of T = is jj f f, where f is the number of standard tableaux of shape ; this formula is given by oefsmit in []. Dene V = as the complex vector space with orthonormal basis fv t j t 2 T = g. Notice that V = is a subspace of V and p t V = V =, where V has basis indexed by standard tableaux of shape. Observation: Let ` n + f and V be an irreducible n+f (q) module. Then p t V is an irreducible p t n+f (q)p t module. This observation implies that there is a set of irreducible representations of p t n+f (q)p t indexed by Young diagrams with n + f boxes, which contain. It follows from equation (9) that M p t V = V 0 = (0) p t n+f? (q)p t where 0 has n + f? boxes and p t V 0 = V 0 =. 0

13 Weights of Markov Traces on ecke Algebras 3 We let z denote the minimal central idempotents of n+f (q), and z = denote the minimal central idempotents of p t n+f (q)p t. Notice that z = has rank equal to the number of skew standard tableaux of shape =. Theorem 3.3 Let f; n be as in Lemma 3.2 and assume that q is not a root of unity. Then f;n as dened in Lemma 3.2 is an onto homomorphism. Proof: In Lemma 3.2 we showed that f;n is a homomorphism. Thus it only remains to show that it is onto. The proof is by induction on n. For n =, we have f; : (q;?q r +m )! p t f + (q)p t. Since ` f is a rectangular diagram there are only two Young diagrams with f + boxes which contain, i.e. [m+; m r? ] and [m r ; ]. As we showed in Lemma 3.2 the action of f; (t) on the representation indexed by [m + ; m r? ] (resp. [m r ; ]) is?qr +m (resp.?). And both representations are dimensional. The algebra (q;?q r +m ) has two irreducible representations indexed by ([]; ;) and (;; []). Both of these representations are dimensional and t 2 (q;?q r +m ) acts by a scalar on these representations. The action of t on V ([];;) (resp. V (;;[]) ) is?q r +m (resp.?). Since q is not a root of unity, these representations are irreducible and nonequivalent. This shows that ([];;) = [m+;m r? ] and (;;[]) = [m r ;]. Assume that for n > we have f;n is onto. Then for all ` n+f containing, V = is an irreducible n (q;?q r +m ), and if ` n+f + is such that, we have V = n(q;?q r +m ) = L 0 V 0 =, as in equation (0). This implies V = is an n+ (q;?q r +m )-module. The remainder of this proof is similar to the proof of irreducibility of modules of the ecke algebra of type A in [W], Theorem 2.2. Let ` n + f + be as described above. By the induction assumption, n (q;?q r +m ) is a semisimple algebra with minimal central idempotents z 0 =. Let 0 6= W V = be an n+ (q;?q r +m ) module. But V = decomposes as an n (q;?q r +m ) module into the direct sum of irreducible modules V 0 = since f;n is onto p t n+f (q)p t. Thus, there exists a 0 ` n + f such that V 0 = W. Let ~ 0 6= 0 be another Young diagram with n + f boxes such that ~ 0. There is exactly one 00 ` n + f? contained in both 0 and ~ 0 such that 00 contains. Let t 2 T = be such that (t ) 0 2 T 0 = and (t ) 00 2 T 00 =. Then (~g n+f (t )) 0 2 T ~ 0 = and therefore (z ~ 0 =) (~g n+f )v t = c d v ~gn+f (t ) 2 V ~ 0 = where d is the axial distance in t between n + f and n + f +. Since q is not a root of unity then c d is well-dened and nonzero, see pg. 0 for the denition of c d. ence the irreducible n (q;?q r +m ) module, V ~ 0 =, is contained in W. But ~ 0 was arbitrary, therefore W L 0 V 0. Next we show that the V = are mutually nonisomorphic n+ (q;?q r +m ) modules. As we observed above this is true for n =. For n = 2 there are ve irreducible modules; 4 are

14 4 R. C. Orellana one dimensional and is two dimensional. We must check that the one dimensional modules are nonequivalent. By the denition of the action of t and ~g f + we have that t acts by?q r +m on V [m+2;m r? ] and V [m+;m+;m r ]. But ~g?2 f + acts by q on V [m+2;m r? ] and by? on V [m+;m+;m r ]. In a similar way we can show that V?2 [m r ;2] and V [m r ; 2 ] are nonequivalent. And since t acts by? on these last two modules, we have that they are nonequivalent to the former two. If n > 2 and and ~ are two distinct partitions of n+f + which contain, then there exists a 0 such that 0 but 0 6 ~. The proof of this fact is found in [W] Lemma 2.. We would also like to remark that restricting to Young diagrams containing does not aect the result. ence, V = and V ~= dier already as n (q;?q r +m ) modules. Remark: Notice that this theorem formalizes the idea we indicated in Section about associating pairs of Young diagrams (; ) with one Young diagram where we adjoin the m r rectangle, see Figure 2. It is well-known that there exits a duality between the quantum group U q (sl(r)) and the ecke algebra of type A. This duality is the quantum analogue of the Schur-Weyl duality between the general linear group, GL(n), and the symmetric group, S n, (see [D], [Ji], and [Ji2]). The following is an easy Corollary of Theorem 3.3. Corollary 3.4 The action of the specialized ecke algebra of type B, n (q;?q r +m ) and the diagonal action of U q (sl(r)) on V V n have the double centralizing property in nd(v V n ). The proof of this corollary follows immeditely from the duality between U q (sl(r)) and the ecke algebra of type A. Remark: To relate the above to the literature we make the following remark. owever, this remark will not be used in the sequel. For denitions we refer the reader to [Ji]. The results of this section imply that there is an R-matrix representation of n (q;?q r +m ) on V V n, where V is the fundamental module of U q (sl(r)) and V is the irreducible module corresponding to. In this R-matrix representation t acts on V V = V V ( and as dened in the proof of Lemma 3.2) as a scalar. And g i! R i is given by the same R-matrices as for the ecke algebra of type A where R i acts on the i-th and i + -th copies of V. 4 Markov Traces on the ecke Algebra of type B In this section we give the necessary background on Markov traces. We refer the reader to [GL] or [G] for the details. A trace function on (q; Q) is a C (q; Q)-linear map : (q; Q)?! C (q; Q) such that (hh 0 ) = (h 0 h) for all h; h 0 2 (q; Q). This denition is in fact valid for any associative algebra over a commutative ground ring. In the case of the group algebra it is clear that

15 Weights of Markov Traces on ecke Algebras 5 every trace function is constant on the conjugacy classes of the underlying group. Notice that (hh 0 )? (h 0 h) = 0; this means that (hh 0? h 0 h) = ([h; h 0 ]) = 0, which implies that the commutators are in the kernel of a trace function in an algebra. The weights we are going to give in this paper correspond to a trace that satises the following denition. Let z 2 C (q; Q) and tr : (q; Q)?! C (q; Q) be an C (q; Q)-linear map. Denition: Then tr is called a Markov trace (with parameter z) if the following conditions are satised: () tr is a trace function on (q; Q); (2) tr() = (normalization); (3) tr(hg n ) = ztr(h) for all n and h 2 n (q; Q). The name of these traces comes from their invariance under the Markov moves for closed braids. Remember that the ecke Algebra is a quotient of the braid group. We note that all generators g i (for i = ; 2; : : : ) are conjugate in (q; Q). In particular, any trace function on (q; Q) must have the same value on these elements. This explains why the parameter z is independent of n in rule (3) of this denition. Geck and Pfeier in [GP] showed that a trace function on the ecke algebra is uniquely determined by its value on basis elements corresponding to representatives of minimal length in the various conjugacy classes of the underlying Coxeter group. Also representatives of minimal length in the classes of Coxeter groups of classical types are of the form d : : : d n, where d i is a distinguished double coset representative of the i (q; Q) with respect to i? (q; Q). Let tr be a Markov trace with parameter z, and let d i 2 D i for i = ; : : : ; n. Then tr(d : : : d n ) = z a tr(t 0 0t 0 : : : t 0 b?) where a is the number of factors d 0 i which are equal to g i? and b is the number of factors which are equal to t 0 i?. Thus, tr is uniquely determined by its parameter z and the values on the elements in the set ft 0 0t 0 : : : t 0 i? j i = ; 2; : : : g. Conversely, given z; y ; y 2 ; : : : 2 C (q; Q) then there exist a unique Markov trace on (q; Q) such that tr(t 0 0 t0 : : : t0 k? ) = y k for all k. For details on these results see [GL], Theorem 4.3. We are particularly interested in the special case when y i = y i for all i 2 N, this is the case when there is only two parameters. In this case, we have that if d i is a distinguished double coset representative then tr(d i x) = tr(x) where = y or z. The proof of the following proposition is found in [GL]. Proposition 4. Let z; y 2 C (q; Q) and tr : (q; Q)?! C (q; Q) be a Markov trace with parameter z such that tr(t 0 0 t0 : : : t0 k? ) = yk for all k then tr(ht 0 n;0) = ytr(h) for all n 0 and h 2 n (q; Q) where t 0 n;0 = g n : : : g tg? : : : g? n or g? n : : : g? tg : : : g n. Notice that the converse is trivially true. A proof of the existence of such traces for every z 2 C is given in [GL]. We will compute the weight vectors for this Markov trace on n (q; Q).

16 6 R. C. Orellana 5 The Weight Formula In this section we dene for every pair of partitions, (; ), a rational function in q and Q, W (;) (q; Q). We will show that this function gives the weights for the Markov trace dened by Geck and Lambropolou [GL] for the ecke algebra of type B. If we denote the weights by! (;) then the Markov trace, tr, can be written as follows: tr(x) = X (;)`n! (;) (;) (x); () where x 2 n (q; Q) and (;) is the character (the usual trace) of the irreducible representation of n (q; Q) indexed by (; ). Let r ; r 2 2 N. First we dene a rational function in q and Q for any arbitrary double partition (; ) such that l() r and l() r 2. If l() = s < r then i = 0 for i = s + ; : : : r, similarly for. Let r = r + r 2. W (;) (q; Q) = q n()+n() (? q Y?? q r )jj+jj +j?i Y? qi?j +j?i? q j?i? q j?i i<jr i<jr 2 Y Yr r 2 i= j= Qq i?i + q j?j Qq?i + q?j (2) Notice that this function can be expressed as a product of Schur functions W (;) (q; Q) = q r jj s (; q; : : : ; q r? )s (; q; : : : ; q r 2? ) s [] (; q; : : : ; q r +r 2? ) jj+jj Y Yr r 2 i= j= ( + Qq i? j +j?i ) : ( + Qq j?i ) (3) Q Recall that s (; q; : : : ; q r? ) = i<jr ned in Section, see equation (). From (3) we can see that W (;) (q; Q) = 0 if l() > r or l() > r 2. Observation: Let?r s r 2?. Assume that q is not a root of unity and Q 6=?q?s. Then W (;) (q; Q) is an analytic rational function.?q i? j +j?i?q j?i is the symmetric Schur function de- The rectangular Young diagram with r rows which has m boxes in the rst r rows and 0 boxes in the remaining rows, i.e. [m r ], has Schur function given by s [m r ];r (q) = qmr (r?)=2 Q r i= Q r2?q m+r +j?i j=?q r +j?i s [] (; q; : : : ; q r? ) mr We are going to assume that for a xed positive integer n and m; r 2 N, we have m > n and r > n. Then for any double partition of n, (; ), we have = [m + ; : : : ; m + :

17 Weights of Markov Traces on ecke Algebras 7 r ; ; : : : ; r2 ] is a partition of n + mr. Then by equation (3) from Section we have the following equation s ;r (q) s [m r ];r (q) = qr jj s (; q; : : : ; q r? )s (; q; : : : q r2? ) s [] (; q; : : : ; q r ) jj+jj Observe that we have the following equality: r Y r 2 Y i= j=? q m+r + i? j +j?i :? q m+r +j?i s ;r (q) s [m r ];r (q) = W (;)(q;?q r +m ): (4) Notice that W (;) (q;?q r +m ) is well-dened since r + m > r always, and as we observed before W (;) (q; Q) is undened for Q =?q?s where?r < s < r 2?. So W (;) (q;?q r +m ) is an analytic rational function. Lemma 5. W (;) (q; Q) = X (;)(;) W (;) (q; Q) where (; ) (; ) means that (; ) is obtained by adding one box to (; ). Proof: Assume = [m + ; : : : ; m + r ; ; : : : ; r2 ]. By the Littlewood-Richardson rule for Schur functions (see [M]) we have the following: s [];r s ;r = X ; jj=jj+ s ;r where s [];r (q) =. equation (4). Now divide both sides of this equation by s [m r ];r (q) and we get by W (;) (q;?q r +m ) = X (;)(;) W (;) (q;?q r +m ): (5) Since W (;) (q; Q) is an analytic rational function and the above equation holds for all Q =?q r +m, thus we have that holds for all values of Q. W (;) (q; Q) = X (;)(;) W (;) (q; Q): (6) In [W] Wenzl showed that the weights of the Markov trace (with parameter z = q r (?q) (?q r ) ) on the ecke algebra of type A are given by the symmetric Schur function s ;r (q) as described in Section.

18 8 R. C. Orellana Let = [m r ] ` f. Since we assumed that m > n and r > n, then we have that for all ` n + f we have that = can be interpreted as a double partition (; ) of n. Now we x t a standard tableau of shape. Then the reduced algebra p t n+f (q)p t has p t as the identity. The Markov trace for the reduced algebra is given by the renormalized Markov trace of n+f (q). By renormalization we mean that we must divide the trace by the s;r (q) trace of the identity, i.e. tr(p t ) = s ;r (q). Therefore, we have that s ;r are the weights (q) of p t n+f (q)p t. Notice that this implies that W (;) (q; Q) reduce to the weights for the reduced algebra when Q =?q r +m. Lemma 5.2 Let g n? 2 n (q; Q), z = qr (?q) (?q r ) and W (;) (q; Q) as dened in equation (2). Then for any x 2 n (q; Q) tr(x) = X (;)`n W (;) (q; Q) (;) (x) (7) denes a well-dened trace which satises the Markov property, i.e. where h 2 n? (q; Q). tr(hg n? ) = ztr(h), Proof: It is clear that tr is indeed a trace. We must show that it satises the Markov property. We have assumed that = = (; ), then W (;) (q;?q r+m ) =. We also s;r (q) s ;r (q) have that = = (;) Q=?q r +m since we have shown in section 3 that V = is an irreducible module of n (q;?q r +m ). Then we have X `n+f s ;r (q) s ;r (q) = (x) = X (;)`n W (;) (q;?q r +m ) (;) (x) denes a Markov trace for the reduced algebra p t n+f (q)p t with parameter z = qr (?q). But (?q r ) in Theorem 3.3 we showed that f;n : n (q;?q r +m )! p t n+f (q)p t is an onto homomorphism. In particular we showed that the irreducible modules of p t n+f (q)p t are irreducible modules for n (q;?q r +m ). Therefore, we have that the above equation also denes a trace which satises the Markov property for n (q;?q r +m ). We know that W (;) (q; Q) and (;) are analytic functions; thus by the identity theorem in complex analysis, since the weights work for all Q =?q r+m, they must work for all Q. Lemma 5. and Lemma 5.2 imply that the function W (;) (q; Q) dened in equation (2) is a weight function for a Markov trace with parameter z = q r (? q)=(? q r ). In Section 4 we noted that a Markov trace on the ecke algebra of type B is uniquely determined by a parameter z and by the values on the set ft 0 0 t0 : : : t0 k? j k g. Therefore, we must still compute the values of the tr on this set in order to completely determine this trace. To compute these values, we need the following denitions.

19 Weights of Markov Traces on ecke Algebras 9 Let A B be a pair of semisimple nite algebras. Given a trace, tr, nondegenerate on both A and B, the conditional expectation A : B! A is dened by setting A (b) to be the unique element of A such that tr( A (b)a) = tr(ba) for all a 2 A and b 2 B A is well-dened and unique. Let B be represented via left multiplication on itself, where we write L 2 (B; tr) to denote the representation space B to distinguish it from the algebra B. We use b to denote the elements in L 2 (B; tr). One obtains from A an idempotent e A : L 2 (B; tr)! L 2 (B; tr) dened by e A b = A (b). The idempotent e A can be thought of as an orthogonal projection onto A with respect to the bilinear form (b ; c )! tr(bc). The algebra hb; e A i generated by B, acting via left multiplication on L 2 (B; tr), and by the idempotent e A is called the Jones basic construction for A B. For the proof of the following results see [J]. Theorem 5.3 Let A, B, A, e A, and tr be as dened above. Then (a) The algebra hb; e A i is isomorphic to the centralizer nd A B of A acting by left multiplication on B. In particular, it is semisimple. (b) There is a - correspondence between the simple components of A and nd A B such that if p 2 A i is a minimal idempotent, pe A is a minimal idempotent of hb; e A i i. (c) e A be A = A (b)e A for all b 2 B. In our case we have the pair of semisimple algebras n+f? (q) n+f (q). The corresponding orthogonal projection is p [ r ], where r is the length of the partitions indexing the irreducible representations. It might also be helpful to keep the following gure in mind, since it clearly shows the veracity of the next lemma. 2 f f+ f+n- f+n f+n+r- 2 r- r r+ r+f r+f+ r+f+n- p [ r ] p [ r ] p t ((((((((( XXX X = p t PP ((((( p t h p [ r ] p [ r ] p t h Figure 7 Before stating the next lemma, we would like to dene a tensor product of ecke algebras given in [GW] by Goodman and Wenzl. We have n (q) m (q) n+m (q) dened by a b = a(shift n (b)) for a 2 n (q) and b 2 m (q), where shift n is the operator which

20 20 R. C. Orellana sends g i to g i+n for all i. In particular, this tensor product allows us to multiply minimal idempotents using the generalized Littlewood-Richardson rule in [GW]. Denote by n the identity in n (q). Lemma 5.4 A Markov trace on the ecke Algebra of type A induces a Markov trace on the ecke algebra of type B, which satises the condition that tr(t 0 nx) = ytr(x), where x 2 n (q; Q). In particular, this implies that y k = y k for all k 2 N. Proof: Let tr be a Markov trace for n+f (q) and tr pt be the Markov trace corresponding to the reduced algebra. We have shown that the weights for the reduced algebra dene a Markov trace for n (q;?q r +m ). Assume that n+f? (q) : n+f (q)! n+f? (q) is the unique conditional expectation with respect to the Markov trace dened by the weight function. Thus, n+f? (q)(~g n+f h) = zh for all h 2 n+f? (q) and z = q r (?q). (?q r ) We denote the image of t 0 n under the homomorphism n+f by p t n+f. Consider the element p t n+f h where h 2 n+f? (q). We would like to compute the conditional expectation of p t n+f h. Observe that n+f? (q)(p t n+f h) = n+f? (q)(p t n+f )h: Therefore it suces to compute the conditional expectation for p t n+f. Consider the following (p t n? p [ r ] )( n+f r? )(p t n? p [ r ] )(h r ) The above expresion is equal to the left hand side of the following equation (see gure 7). [((p [ r ] p t )( r? n+f )(p [ r ] p t )) n? ]( r h) = (const)(p [ r ] p t n? )( r h) Since we are restricted to r rows we have by the generalization of the Littlewood-Richardson rule, see [GW], that p [ r ] p t is a minimal idempotent. But Theorem 5.3 (c) implies that (const)p t = n+f? (q)( n+f p t ). Thus, tr pt ( n+f h) = (const)tr pt (h): This implies that we have a Markov trace with the property tr(t 0 nh) = (const)tr(h) for all h 2 n+f? (q). But this trace is also a trace for n (q;?q r +m ). Since the above property holds for all Q =?q r +m, then it must hold for all Q. Thus our assertion is proved. Theorem 5.5 Let r ; r 2 2 N and set r = r + r 2. If tr is a Markov trace on the ecke algebra of type B, with parameter z = q r (? q)=(? q r ), such that tr(t 0 0 t0 : : : ; t0 ) = k? yk for k. Then the weights are given by W (;) (q; Q) as dened in equation (2) with y = (q r 2 Q + )(? qr )=(? qr )?.

21 Weights of Markov Traces on ecke Algebras 2 Proof: The fact that the weights dene a Markov trace follows from Lemmas 5., 5.2 and 5.4. It remains to show that the weight formula is indeed given by this values of y = (Qq r 2 + )(? qr )=(? qr +r 2 )?. By Lemma 5. it suces to compute tr(t) in (q). This is a straight forward computation, we have W ([];;) (q; Q) = (?qr )(+Qq r 2 ) (?q r )(+Q) and W (;;[]) (q; Q) = qr (?q r )(+Qq?r ). Also ([];;) (t) = Q and (;;[]) (t) =?. Using these values (?q r )(+Q) we compute tr(t) = QW ([];;) (q; Q)? W (;;[]) (q; Q) = (Qqr 2 + )(? qr ) (? q r )? : 6 Markov Trace for the ecke algebra of type D The easiest way to study Markov traces on the ecke algebras of type D is by embedding these algebras into those for type B, and then applying the results for type B. As observed by oefsmit [], in order to obtain an embedding of D n (q) into B n (q; Q) we have to set the parameter Q equal to. Notice that in this case we have t 2 =. The ecke algebra of type D is generated by u = tg t; g ; : : : ; g n? satisfying the following relations: (D) g i g i+ g i = g i+ g i g i+ for i = ; : : : n? 2; (D2) g i g j = g j g i whenever ji? jj 2; (D3) g 2 i = (q? )g i + q for all i; (D4) g i u = ug i for all i; (D5) u 2 = (q? )u + q. We have D n (q) B n (q; Q) for all n; then D (q) = S n> D n (q) B (q; Q). Geck in [G] shows that the restriction of a Markov trace on B (q; Q) is a Markov trace on D (q) and both have the same parameter. Furthermore, he shows that every Markov trace on D (q) can be obtained in this way. From oefsmit [] we know that the simple components for D n (q) are indexed by double partitions (; ). If 6= we have that the B n (q; Q)-modules V (;) and V (;) are simple, equivalent D n (q)-modules. And if = we have that the B n (q; Q)-module V (;) decomposes into two simple nonequivalent D n (q)-modules, i.e. V (;) i with i = ; 2. Using Bratteli diagrams we have the following relations for simple modules of the B n (q; Q) and D n (q)

22 22 R. C. Orellana type B type D 6= (; ) (; ) (; ) = (; ) (; ) (; )2 gure 8 Proposition 6. Let r ; r 2 2 N and set r = r +r 2. Then the weight formula for the Markov trace on the ecke algebra of type D with parameters z = q r (?q) and y = (Qqr 2 +)(?q r )? (?q r ) (?q r ) is given as follows: W D (;)(q) = W (;) (q; ) + W (;) (q; ); if 6= and W D (;) i (q) = W (;) (q; ); for i = ; 2 if = where W (;) (q; ) denote the weight evaluated at Q = of the ecke algebra of type B. Proof: The proof of this theorem follows directly from the inclusion matrix of the ecke algebra of type D into the ecke algebra of type B. Recall that in order to obtain the weight vector for the ecke algebra of type D we multiply the inclusion matrix for n D (q) n B (q; Q) with the weight vector for type B. References [B] Birman, J, Braids, links and mapping class groups, Ann. Math. Stud. 82 (974). [D] Drinfeld, V. Quantum groups, Proceedings ICM Berkeley, 986, [DJ] Dipper, R.; James, G.D, Representations of ecke algebras of type B n, J. Algebra 46 (992), [G] Geck, Meinolf, Trace functions on Iwahori-ecke Algebras, Banach Center Publications 42 (998), [GL] Geck, Meinolf; Lambropoulou, Soa, Markov traces and Knot invariants related to Iwahori-ecke algebras of type B, J. Reine Angew. Math , [GP] Geck, Meinolf; Pfeier, On the irreducible characters of ecke algebras, advances in Math. 02 (993),

23 Weights of Markov Traces on ecke Algebras 23 [GW] Goodman, F. M.; Wenzl,. Littlewood-Richardson Coecients for ecke Algebras at Roots of Unity, Advances in Mathematics. 82, No2, August (990). [] oefsmit, P.N., Representations of ecke algebras of nite groups with BN-pairs of classical type, thesis, University of British Columbia, 974. [Ji] Jimbo, M., Quantum R-matrices for generalized Toda System, Comm. Math. Phys., 02, 4(986), [Ji2] Jimbo, M., A q-analogue of U(gl(N+)), ecke Algebras and the Yang-Baxter equation, Lett. Math. Phys., 0 (985), [J] Jones, V.F.R., Index for Subfactors, Invent. Math 72 (983) -25. [J2] Jones, V.F.R., ecke algebra representations of braid groups and link polynomials, Ann of Math, 26 (987), [K] Kassel, C. Quantum Groups, Springer-Verlag, New York, (995). [M] Macdonald, I, Symmetric Polynomials and all Polynomials, sec. ed., Clarendon Press, Oxford, (995) [RW] Ram, A.; Wenzl,., Matrix units for centralizer algebras, Journal of Algebra, 45, No.2 992), [W] Wenzl,. ecke algebras of type A n and subfactors, Invent. math 92, (988). [W2] Wenzl,. Braids and Invariants of 3-manifolds, Invent. math. 4, (993). [Y] Young, A. On quantitative Substitutional Analysis V. Proceedings of the London Mathematical Society, 3(2) (930) University of California, San Diego, Department of Mathematics, La Jolla, CA 9306, USA -mail address: rorellan@math.ucsd.edu