generic algebras and kazhdan-lusztig theory for monomial groups

APPROVED: J. Matthew Douglass, Major Professor Elizabeth Bator, Committee Member Douglas Brozovic, Committee Member Anne Shepler, Committee Member Nathaniel Thiem, Committee Member Neal Brand, Chair of the Department of Mathematics Sandra L. Terrell, Dean of the Robert B. Toulouse School of Graduate Studies GENERIC ALGEBRAS AND KAZHDAN-LUSZTIG THEORY FOR MONOMIAL GROUPS Shemsi I. Alhaddad Dissertation Prepared for the Degree of DOCTOR OF PHILOSOPHY UNIVERSITY OF NORTH TEXAS May 2006 UMI Number: 3214450 3214450 2006 UMI Microform Copyright All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. Alhaddad, Shemsi I. Generic Algebras and Kazhdan-Lusztig Theory for Monomial Groups. Doctor of Philosophy (Mathematics), May 2006, 49 pp., references, 6 titles. The Iwahori-Hecke algebras of Coxeter groups play a central role in the study of representations of semisimple Lie-type groups. An important tool is the combinatorial approach to representations of Iwahori-Hecke algebras introduced by Kazhdan and Lusztig in 1979. In this dissertation, I discuss a generalization of the Iwahori-Hecke algebra of the symmetric group that is instead based on the complex reflection group G(r,1,n). Using the analogues of Kazhdan and Lusztig's R-polynomials, I show that this algebra determines a partial order on G(r,1,n) that generalizes the Chevalley-Bruhat order on the symmetric group. I also consider possible analogues of Kazhdan-Lusztig polynomials. ii ACKNOWLEDGMENTS I would like to thank my cat Joe for not eating the only extant copy of my dissertation at the same time as he gnawed through all my library books. My deepest apologies to the UNT library. I am very grateful to Matt Douglass for his constant support and encouragement, without which none of this would have been possible. I am very fortunate to have had a colleague like Sam Tajima, who raised interesting questions and whose enthusiasm for mathematics is inspiring. I would also like to thank my committee members for their time and all of their wonderful advice and moral support. On a personal note, I am grateful to Helen-Marie, Joe (my brother, not the cat), Woody, Robin and Cely for feigning interest in my work, and keeping me in touch with the world outside the math department. My deepest appreciation goes to Andy, for his love and patience, and the fact that he is looking over my shoulder as I type this. This thesis is dedicated in memory of Yuba. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS ii Chapters 1. INTRODUCTION 1 2. INDUCED MODULES AND THE HECKE ALGEBRA 3 3. AN EXAMPLE OF A HECKE ALGEBRA 9 4. THE GENERIC HECKE ALGEBRA 19 5. A PARTIAL ORDER ON W H b 27 6. THE R-POLYNOMIALS 40 BIBLIOGRAPHY 49 CHAPTER 1 INTRODUCTION In the 1800s, groups were typically regarded as subsets of the p ermutations of a set, or of GL(V ) where V is a vector space. In this past century, abstract groups were introduced, and group theory split into two branches: the study of abstract groups, and the study of the ways in which a group may be embedded in GL(V ). Representation theory entails the latter. Representation theory is used as a conduit between abstract groups and real world applications, including robotics, telephone network and stereo designs, and models for elementary particles. In the past few decades, combinatorial representation theory has grown to prominence. Combinatorial representation theory gives group theorists explicit tools for dealing with abstract concepts. This thesis takes a combinatorial approach to the study of a specific type of Hecke algebra. The second chapter of this thesis contains background information. In this chapter we define induced modules and an algebra called the Hecke algebra. In Theorem 2.2 we describe an important relationship between the decomposition of induced modules and the decomposition of Hecke algebras. Afterwards we describe properties of the Hecke algebra, including a depiction of the Hecke algebra as an algebra of functions that are constant on specific cosets. In this chapter we also describe a nice basis for the Hecke algebra and find a simple formula for the structure constants associated with this basis. In Chapter 3 we analyze a specific Hecke algebra H a . Adapting the work of Chapter 2 to H a we define a basis for the algebra and use the structure constants to describe multiplication relations on the basis. In the fourth chapter, we define a Hecke algebra H, as an A-algebra with certain gener- ators and relations, where A is the ring Z[a, q 1/2 , q −1/2 ]. Then we adapt the work of Lusztig [5, Proposition 3.3] to show that H is a free A-module. 1 In Chapter 5 we define a partial order that arises from the basis of H, where H is viewed as a free A-module. This partial order agrees with the Bruhat-Chevalley order if we insist that the indeterminate a is equal to the indeterminate q − 1. This chapter contains many interesting properties of H, including the fact that the basis elements are invertible. Writing the inverse of each basis element as a linear combination of basis elements, we analyze the coefficients of the basis elements in the linear combination and find a recursive definition for the coefficients. In the sixth chapter we adapt the work of Deodhar [3] to show that the coefficients found in Chapter 5 are polynomial in aq −1/2 . 2 CHAPTER 2 INDUCED MODULES AND THE HECKE ALGEBRA In this chapter we define induced modules and an algebra called the Hecke algebra. In Theorem 2.2 we describe an important relationship b e tween the decomposition of induced modules and the decomposition of Hecke algebras. Afterwards we describe properties of the Hecke algebra, including a depiction of the Hecke algebra as an algebra of functions that are constant on specific cosets. In this chapter we also describe a nice basis for the Hecke algebra and find a simple formula for the structure constants associated with this basis. Unless otherwise noted in this chapter, k is a field and V is a vector space over k, G is a finite group and H is a fixed subgroup of G. A k-representation of a group G is a group homomorphism from G to GL(V ). If V is of finite dimension n, it is common to cho ose a basis for V and identify GL(V ) with GL n (k). The k-group algebra, kG, associated to G is the algebra of functions from G to k, with pointwise addition and scalar multiplication, and multiplication given by the convolution product (f · g)(x) =  y∈G f(y)g(y −1 x) for x in G. A typical function in kG looks like  g∈G a g u g , where u g is the characteristic function of {g}. There is an easy way to move betwee n k-representations and kG-modules. Starting with a k-representation ρ : G → GL(V ), we define an action from kG×V to V by   g∈G a g u g  v =  g∈G a g ρ(g)(v). It is easy to check that this action defines a left kG-module structure on V . Therefore, each k-representation ρ : G → GL(V ) gives V a left kG-module structure. On the other hand, starting with a left kG-module V , we define a map T g : V → V by T g (v) = u g v, where g is in G. It is easy to check that T g is an inve rtible linear transformation. 3 Next we define ρ : G → GL(V ) by ρ(g) = T g . It is straightforward to verify that ρ is a k-representation. In this way every left kG-module V determines a k-representation ρ : G → GL(V ). Before getting to the major ideas of this chapter, we need a few more definitions. For a ring R, a nonzero R-module M is said to be simple if M has no proper nonzero submodules. If a nonzero R-module N can be written as a direct sum of simple R-modules, then N is called semisimple. Moreover, a ring R is called semisimple if every R-module is isomorphic to a direct sum of simple R-modules. The next theorem gives conditions under which kG is a semisimple ring. Theorem 2.1 (Maschke’s Theorem). If k is a field, G is a finite group, and the characteristic of k does not divide |G|, then the group algebra kG is a semisimple ring. Proof. See [6, Theorem 8.47]  From now on we assume the characteristic of k does not divide |G|. By Maschke’s Theorem, kG is se misimple, and so kG-modules can be decomposed into a direct sum of simple kG-modules. Therefore, to describe an arbitrary kG-module, it is enough to determine the simple kG-modules. A main theorem in this chapter, Theorem 2.2, uses this idea along with induction. The operation of induction from kH-modules to kG-modules assigns to each kH-module, L, a left kG-module, Ind G H (L), called the induced module, given by Ind G H (L) = kG ⊗ kH L. Now we consider the special case where L has the form kHe, for an idempotent e in kH. In this case, it is easy to see that Ind G H (L) = kG ⊗ kH kHe ∼ = kGe and so End kG (Ind G H (L)) ∼ = End kG (kGe) = Hom kG (kGe, kGe) ∼ = (ekGe) op . The last isomorphism is given by f → f(e), where f is a homomorphism from kGe to kGe. From now on we only consider induced mo dules of the form Ind G H (kHe) where e is an 4 idempotent in kH. In this case, there are strong links b etween the Hecke algebra ekGe and the decomposition of Ind G H (kHe) into irreducible constituents. A k-algebra, A, is said to be split semisimple if A is semisimple and End A (L) = k · 1 L for each simple left A-module L. We assume from now on that kG is split semisimple. Theorem 2.2. Using the notation defined above, we have the following three statements. (i) The algebra ekGe is split semisimple. (ii) The irreducible constituents of Ind G H (L) correspond to the irreducible ekGe-modules. In particular, if M is an irreducible constituent of Ind G H (L), then M corresponds to the irreducible ekGe-module eM . (iii) The multiplicity of each irreducible constituent of Ind G H (L) is equal to the dimension of the corresponding ekGe-module. Proof. To prove the second statement, we define a map by M → eM, and show the map is bijective.  Now consider the very special case when L is the trivial kH-module. Then L = kHe H where e H = |H| −1  h∈H u h . Theorem 2.3. The algebra e H kGe H is the subalgebra of kG that consists of k-valued func- tions on G that are constant on (H, H) double cosets of G. Proof. To show that every element of e H kGe H is constant on (H, H) double cosets, it is enough to show that the basis elements of e H kGe H are constant on double cosets. In particular, consider e H u g e H where g is in G. Fix x in G. Then using the convolution product twice, we get (e H u g e H )(x) = (e H (u g e H ))(x) =  y∈G e H (y)(u g e H )(y −1 x) =  y∈G e H (y)  z∈G u g (z)e H (z −1 y −1 x) 5 [...]... )), for s in S and π in U, and ˜ ˜ ˜ • Pd (te )π(te ) = Pd (π(te )), for d in Hb and π in U ˜˜ ˜ ˜˜ ˜ Therefore, ts tx = Ps (tx ), for s in S and x in W Hb , and td tx = Pd (tx ), for d in Hb and x in W Hb Thus, (3) (4) ˜˜ ˜ ts tx = tsx , if (sw) > (w) ˜˜ ˜ ts tx = tsx + aq −1/2 ˜ tdx , if (sw) < (w) d∈Xs (5) ˜˜ ˜ td tx = tdx for d ∈ Hb 25 ˜ ˜ ˜ ˜ From statement (3), we see tsi tsj = tsj tsi for si and. .. ALGEBRA In this and all remaining chapters, assume a and q are indeterminates Let A denote Z[a, q 1/2 , q −1/2 ] Define H to be the A-algebra with generators { ts , td | s ∈ S, d ∈ Hb } and relations: (i) td td = tdd for d and d in Hb , (ii) td ts = ts tsds for d in Hb and s in S, (iii) tsi tsj = tsj tsi for si and sj in S with |i − j| > 1, (iv) tsi tsi+1 tsi = tsi+1 tsi tsi+1 for si in S and 1 ≤ i < n,... s = v −1 and δ = e Therefore w = sv and δ = e, and since sus is in Ba , we get −1/2 −1/2 1/2 − − µ(s,v,sv) = qs qv qsv |{ u ∈ Us | v −1 sus ∈ Ba v −1 Ba }| = |Us | = 1 17 Now suppose (w−1 s) < (w−1 ) and u = e Then w−1 s is in Ba v −1 δ Ba if and only if w−1 s = v −1 δ This in turn is true if and only if w−1 s = v −1 and δ = e If w−1 s = v −1 and (w−1 s) < (w−1 ) then (v −1 ) < (v −1 s), and this... tsi+1 , for si in S and 1 ≤ i < n, (v) ts ts = te + aq −1/2 δ∈Xs tδs , for s in S Recall that if si1 sip is a reduced expression for w in W , then tw = tsi1 tsip Also, twd = tw td and tdw = td tw , for w ∈ W and d ∈ Hb , and te is the identity element in H Since Hb is a group, each d in Hb has an inverse, d−1 Using relation (i), we see that td td−1 = tdd−1 = te Therefore, t−1 = td−1 for each... For x and y in W Hb , define Rx,y in A by ty = Rx,y tx x∈W Hb Since td = td for d in Hb , we see that Rx,d = 1 if x = d and is zero otherwise Since ts = ts − aq −1/2 δ∈Xs tδ for s in S, we have Rx,s = 1 if x = s, Rx,s = aq −1/2 if x is in Xs , and Rx,s = 0 otherwise For x and y in W Hb , we write x chapter is that y if and only if Rx,y = 0 The main result of this defines a partial order on W Hb Before... and so te = ts ts − aq −1/2 tsδ = ts ts − aq −1/2 δ∈Xs ts tδ = ts δ∈Xs 27 ts − aq −1/2 tδ δ∈Xs Therefore, t−1 = ts − aq −1/2 s δ∈Xs tδ For w in W and d in Hb , we have twd = tw td = tsi1 tsip td , where w = si1 sip is reduced We see that t−1 = t−1 t−1 t−1 Therefore, twd is a unit for each wd in W Hb si1 wd d sip Lemma 5.2 For x in W Hb , d in Hb and s in S we have (6) (7) td tx = tdx and. .. are indeterminates, and A = Z[a, q 1/2 , q −1/2 ] Lemma 5.1 For each x in W Hb , the basis element tx is a unit in H Proof Recall from the previous chapter that H is the A-algebra whose generators are given by { ts , td | s ∈ S, d ∈ Hb } and whose relations are: (i) td td = tdd , for d and d in Hb , (ii) td ts = ts tsds , for d in Hb and s in S, (iii) tsi tsj = tsj tsi , for si and sj in S with |i... set { tx | x ∈ W Hb } Before doing so, we need a few lemmas Lemma 4.3 Suppose s and r are in S, w is in W , (swr) = (w), and (sw) = (wr) Then sw = wr Proof There are two cases First suppose (wr) > (w), and so (wr) > (swr) Suppose by way of contradiction that swr = w, and fix a reduced expression si1 sip for w Since swr = w and (swr) = (w), we have swr = s1 si sp r, for some i with 1 ≤ i ≤ p,... (tx ) = tdx , for d ∈ Hb ; ˜ ˜ Qs (tx ) = txs , if (xs) > (x); ˜ ˜ Qs (tx ) = txs + aq −1/2 ˜ d∈Xs txd , if (xs) < (x); ˜ ˜ Qd (tx ) = txd , for d ∈ Hb Lemma 4.6 Py Qz = Qz Py , for all y and z in S ∪ Hb Proof Fix x is in W Hb The proof of this lemma has nine cases, the first six of which show that Ps Qr = Qr Ps , for s and r in S The seventh case shows that Pd1 Qd2 = Qd2 Pd1 for d1 and d2 in Hb ... By Lemma 4.4 and the fact that rXr r = Xr , we have δ Xr = rδXr r = δXr Therefore, ˜ tswδ d = d∈Xr Hence, ˜ d∈Xs tdxr = ˜ d∈Xr tsxd , ˜ tswδd = d∈Xr ˜ tsxd d∈Xr and so Ps Qr = Qr Ps Case 7: Suppose d1 and d2 are in Hb Then ˜ ˜ ˜ ˜ ˜ Pd1 Qd2 (tx ) = Pd1 (txd2 ) = td1 xd2 = Qd2 (td1 x ) = Qd2 Pd1 (tx ) Therefore, Pd1 Qd2 = Qd2 Pd1 , for d1 and d2 in Hb Case 8: Suppose d is in Hb and r is in S We . Neal Brand, Chair of the Department of Mathematics Sandra L. Terrell, Dean of the Robert B. Toulouse School of Graduate Studies GENERIC ALGEBRAS AND KAZHDAN-LUSZTIG THEORY FOR MONOMIAL GROUPS. Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346 by ProQuest Information and Learning Company. Alhaddad, Shemsi I. Generic Algebras and Kazhdan-Lusztig. Robin and Cely for feigning interest in my work, and keeping me in touch with the world outside the math department. My deepest appreciation goes to Andy, for his love and patience, and the