A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....b B n have many important applications to the low dimensional topology, see e.g. B N] Dr2] F] Ko3] L] In Section 3 we define the Jucys Murphy elements d k in the braid algebra B n , and the multiplicative Jucys Murphy elements D k in the pure braid group P n . Follow to A. Ram, [Ra], we prove that the Jucys Murphy element d k 2 B n is the quasi classical limit of the element D k 2 P n . We prove also, that the infinitesimal deformation of the multiplicative Jucys Murphy element D k coincides with the element d k . In Section 4 we define the quadratic algebra G n and ....

.... words, there exists a homomorphism p from the braid algebra B n to the group ring of the symmetric group, p : B n n ] such that p(X ij ) i; j) Under this homomorphism p the element d j maps to the Jucys Murphy element in the group ring of the symmetric group S n (see e.g. J] Mu] [Ra]) A proof follows from the following relations between the transpositions in the symmetric group S n : if 1 i j k n, then (ij) ik) jk) ij) ik) jk) ij) jk) ik) ij) jk) ik) Theorem 3.3 Let K n be a commutative subalgebra of B n generated by the Jucys Murphy elements d j , ....

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Ram A., Seminormal representations of Weyl groups and Iwahori--Hecke algebras, 1995, to appear in Proc. London Math. Soc.;

....presentation with generators f i g; i = 1; n 1 corresponding to Artin s elementary braids, which satisfy the braid relations and the quadratic relations 2 i = z i 1. The braids T (j) j = 1; n, shown here make up a set of commuting elements in H n . These are shown by Ram [9] to be the Murphy operators of Dipper and James [2] up to linear combination with the identity of H n . T (j) j 2 Their properties are discussed further in [8] where the element T (n) is shown to represent their sum, again up to linear combination with the identity. T (n) In the same ....

A.Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. London Math. Soc. 75 (1997), 99-133.

....factor through C n , it follows that C n and H n (q) are isomorphic. 3.1. Jucys Murphy Elements. Hoefsmit [Ho] defines special elements in H n (q) which act diagonally on the seminormal representations. The analogous elements in S n later became known as Jucys Murphy elements (see [Ra]) We now define analogous elements in I n (q) For 1 # i # n, define X i = q (i 1) T i 1 T i 2 T 1 ) 1 P 1 ) T 1 T 2 T i 1 ) so that X i = q 1 T i 1 X i 1 T i 1 , for i # 2. REPRESENTATIONS OF THE q ROOK MONOID 15 Figure 1. Bratteli Diagram for I n (q) n =0: # . ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3), 75 (1997), 99-133.

....and their sum, nishing with an elegant representation of their power sums and other symmetric polynomials. Starting in section 2 with a choice of n string braids corresponding to the transpositions I exhibit a braid T (j) representing each of the individual Murphy operators M(j) following Ram [15], and then a very natural simple n tangle T (n) which represents their sum M , up to a linear combination with the identity element in H n in each case. From the tangle viewpoint it becomes immediately clear that M is central. The Hom y skein of the annulus, C, has been exploited for many years, ....

....operator M(j) can be represented in H n by a single braid T (j) up to linear combination with the identity. Theorem 2.2 (Morton) The sum M of the Murphy operators can be represented in H n by a single tangle T (n) again up to linear combination with the identity. Proof of theorem 2. 1: In [15] Ram notes that M(j) can almost be represented by the single braid T (j) pictured, where almost means that M(j) is a linear combination of T (j) and the identity. Set T (j) j 5 in H n , for each j n including the case T (1) 1. Skein theory shows quickly that T (j) 1 = zM(j) giving M(j) ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. London Math. Soc. 75 (1997), 99-133.

....for Iwahori Hecke algebras of type A. In 1994 Ariki and Koike [AK] introduced (some of) the cyclotomic Hecke algebras and generalized Hoefsmit s construction to these algebras. The construction was generalized to a larger class of cyclotomic Hecke algebras in [Ar2] For a summary of this work see [Ra1] and [HR] General ane Hecke algebras are naturally associated to a reductive algebraic group and the size of the commutative part of the ane Hecke algebra depends on the structure of the corresponding algebraic group (simply connected, adjoint, etc. The second aim of this paper is to show that ....

....are all special cases of Theorem 3.8. 3.19) Jucys Murphy elements in cyclotomic Hecke algebras. The following result is well known, but we give a new proof which shows that the cyclotomic Hecke algebra analogues of the Jucys Murphy elements which have appeared in the literature (see [BMM] [Ra1], DJM] and the references there) come naturally from the ane Hecke algebra H1;1;n . Corollary 3.20. Let H r;1;n (u 1 ; u r ; q) and H r;p;n (x 0 ; x d 1 ; q) be the cyclotomic Hecke algebras de ned in (1.1) and (1.2) a) The elements M i = T i T 2 T 1 T 2 T i ; 1 ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.

....IRREDUCIBLE REPRESENTATIONS OF THE IWAHORI HECKE ALGEBRA OF TYPE F 4 ARUN RAM AND D. E. TAYLOR Abstract. A general method for computing irreducible representations of Weyl groups and Iwahori Hecke algebras was introduced by the first author in [8]. In that paper the representations of the algebras of types An , Bn , Dn and G 2 were computed and it is the purpose of this paper to extend these computations to F 4 . The main goal here is to compute irreducible representations of the Iwahori Hecke algebra of type F 4 by only using information ....

....(w 0 )p c( s) c( 2.2) c( s) N s (r s ) and c( l) N l (r l ) Let be a realization of the irreducible representation indexed by and let Id be the d Theta d identity matrix, where d is the dimension of . Then we have the following result [7] 5] [8]: a) If w 0 is central in W then (T w 0 ) c( Id ; b) If w 0 is not central in W then w 0 ) c( Id : 3. Seminormal representations We shall compute the irreducible representations of HF 4 inductively: the representations of HA 1 are one dimensional and one can immediately ....

[Article contains additional citation context not shown here]

A. Ram. Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. London Math. Soc. (3), 75:999--999, 1997.

.... that the methods of Section 7 should also produce analogous results for the Weyl groups of type B n and the imprimitive complex re ection groups G(r; 1; n) The long and short systematic scans can be de ned in a similar way and the representation theory goes through without problems (see [Hf] [Ra] and [AK] The remaining necessary ingredient is an analogue of Lemma 7.2. Acknowledgement. This paper is dedicated to our friend Bill Fulton. We are also thankful to Ruth Lawrence for early e orts to help understand deformed random walks. 2. Probabilistic background In this section we give ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.

....IRREDUCIBLE REPRESENTATIONS OF THE IWAHORI HECKE ALGEBRA OF TYPE F 4 ARUN RAM AND D. E. TAYLOR Abstract. A general method for computing irreducible representations of Weyl groups and Iwahori Hecke algebras was introduced by the first author in [8]. In that paper the representations of the algebras of types An , Bn , Dn and G 2 were computed and it is the purpose of this paper to extend these computations to F 4 . The main goal here is to compute irreducible representations of the Iwahori Hecke algebra of type F 4 by only using information ....

....c( 2.2) where c( s) N s (r s ) 1) and c( l) N l (r l ) 1) Let be a realization of the irreducible representation indexed by and let Id be the d Theta d identity matrix, where d is the dimension of . Then we have the following result [7] 5] [8]: a) If w 0 is central in W then (Tw 0 ) c( Id ; b) If w 0 is not central in W then (T 2 w 0 ) c( 2 Id : 3. Seminormal representations We shall compute the irreducible representations of HF 4 inductively: the representations of HA 1 are one dimensional and one can ....

[Article contains additional citation context not shown here]

A. Ram. Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. London Math. Soc. (3), 75:999--999, 1997.

....by explicit formulas which give the action of each generator of the affine Hecke algebra on a specific basis, the elements of which are indexed by standard Young tableaux. This is a generalization of the constructions of A. Young [Y] P. Hoefsmit [Ho] H. Wenzl [Wz] and Ariki and Koike [AK] see [Ra2] for a review of some of the unpublished results of Hoefsmit) Parts of Theorem 4.1 were first discovered by Cherednik and are stated (without proof) in [Ch] I am grateful to A. Zelevinsky for pointing this out to me and to I. Cherednik for some informative discussions. 2) The definition of an ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.

....for Iwahori Hecke algebras of type A. In 1994 Ariki and Koike [AK] introduced (some of) the cyclotomic Hecke algebras and generalized Hoefsmit s construction to these algebras. The construction was generalized to a larger class of cyclotomic Hecke algebras in [Ar2] For a summary of this work see [Ra1] and [HR] General affine Hecke algebras are naturally associated to a reductive algebraic group and the size of the commutative part of the affine Hecke algebra depends on the structure of the corresponding algebraic group (simply connected, adjoint, etc. The second aim of this paper is to show ....

....are all special cases of Theorem 3.8. 3.19) Jucys Murphy elements in cyclotomic Hecke algebras. The following result is well known, but we give a new proof which shows that the cyclotomic Hecke algebra analogues of the Jucys Murphy elements which have appeared in the literature (see [BMM] [Ra1], DJM] and the references there) come naturally from the affine Hecke algebra H1;1;n . Corollary 3.20. Let H r;1;n (u 1 ; u r ; q) and H r;p;n (x 0 ; x d Gamma1 ; q) be the cyclotomic Hecke algebras defined in (1.1) and (1.2) a) The elements M i = T i Delta Delta Delta T 2 ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.

....the generators of the Hecke algebra. Using certain surjective homomorphisms [A] from the affine Hecke algebras of type A to the algebras H r;1;n one can easily show that these earlier constructions are type A special cases of the general type construction given in this paper. In a previous paper [Ra1] I gave a method for generalizing Young s theory of seminormal representations to general Lie type. I now believe that this earlier idea was not the proper way to proceed. The method here is much more natural and yields a cleaner and more beautiful theory. Young s classical formulas for the ....

A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.

....IRREDUCIBLE REPRESENTATIONS OF THE IWAHORI HECKE ALGEBRA OF TYPE F 4 ARUN RAM AND D. E. TAYLOR Abstract. A general method for computing irreducible representations of Weyl groups and Iwahori Hecke algebras was introduced by the first author in [8]. In that paper the representations of the algebras of types An , Bn , Dn and G 2 were computed and it is the purpose of this paper to extend these computations to F 4 . The main goal here is to compute irreducible representations of the Iwahori Hecke algebra of type F 4 by only using information ....

....(2.2) where c( s) N s (r s ) 1) and c( l) N l (r l ) 1) Let be a realization of the irreducible representation indexed by and let Id be the d Theta d identity matrix, where d is the dimension of . Then we have the following result [7] 5] [8]: a) If w 0 is central in W then (T w 0 ) c( Id ; b) If w 0 is not central in W then (T 2 w 0 ) c( 2 Id : 3. Seminormal representations We shall compute the irreducible representations of HF 4 inductively: the representations of HA 1 are one dimensional and one can ....

[Article contains additional citation context not shown here]

A. Ram. Seminormal representations of Weyl groups and Iwahori-Hecke algebras. Proc. London Math. Soc. (3), 75:999--999, 1997.

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A. Ram, Seminormal representations of Weyl groups and Iwahori-Hecke algebras, Proc. London Math. Soc. (3) 75 (1997), 99-133.

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