Journal of Algebra 323 (2010) Contents lists available at ScienceDirect. Journal of Algebra.

Transcription

1 Jounal of Algeba 33 (00) Contents lists available at ScienceDiect Jounal of Algeba Paametes fo which the Lawence Kamme epesentation is educible Claie Levaillant, David Wales Caltech, MC 53-37, Pasadena, CA 95, United States aticle info abstact Aticle histoy: Received 3 August 008 Available online Febuay 00 Communicated by Michel Boué Keywods: Repesentation theoy Lawence Kamme epesentation Baid goups BMW algebas Semisimplicity Iwahoi Hecke algebas We show that the epesentation, intoduced by Lawence and Kamme to show the lineaity of the baid goup, is geneically ieducible. Howeve, fo some values of its two paametes when these ae specialized to complex numbes, it becomes educible. We constuct a epesentation of degee n(n ) of the BMW algeba of type A n. As a epesentation of the baid goup on n stands, it is equivalent to the Lawence Kamme epesentation whee the two paametes of the BMW algeba ae elated to those appeaing in the Lawence Kamme epesentation. We give the values of the paametes fo which the epesentation is educible and give the pope invaiant subspaces in some cases. We use this epesentation to show that fo these special values of the paametes, the BMW algeba of type A n is not semisimple. 00 Elsevie Inc. All ights eseved.. Intoduction.. Intoduction and main esults In [8], Daan Kamme constucted a faithful linea epesentation of the baid goup. Since this epesentation was ealie intoduced by Ruth Lawence in [9], it is called the Lawence Kamme epesentation. Stephen Bigelow uses this same epesentation in [] to show independently fom Kamme that the baid goup is linea. A genealization of the lineaity esult fo the baid goup to the othe Atin goups of finite type is given in [5] and independently in [6]. In this pape, we examine a epesentation of degee n(n ) of the BMW algeba of type A n. As a epesentation of the baid goup on n stands, this epesentation is equivalent to the Lawence Kamme epe- * Coesponding autho. addesses: cl@caltech.edu (C. Levaillant), dbw@caltech.edu (D. Wales) /$ see font matte 00 Elsevie Inc. All ights eseved. doi:0.06/j.jalgeba

2 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) sentation (abbeviated L K epesentation). By studying this epesentation we show that the L K epesentation is geneically ieducible. Howeve, fo some values of its two paametes when these ae specialized to complex numbes, it becomes educible. Thoughout the pape, we let l, m and be thee nonzeo complex numbes, whee m and ae elated by m =. WedefineH F,(n) as the Iwahoi Hecke algeba of the symmetic goup Sym(n) ove the field F = Q(l, ) with geneatos g,...,g n. They satisfy the baid elations and the quadatic elation g i + mg i = foalli. Ou definition is the same as the definition of [] afte the geneatos have been escaled by a facto. Ou main esult is as follows. Theoem (Main theoem). Let n be an intege with n 3 and let m, l and be thee nonzeo complex numbes, whee m and ae elated by m =. Assume that H F,(n) is semisimple, and so assume that k fo evey intege k {,...,n}. When n 4, the Lawence Kamme epesentation of the BMW algeba of type A n with paametes l and m ove the field Q(l, ) is ieducible, except when l {, 3,,, },whenitis n 3 n 3 n 3 educible. When n = 3, the Lawence Kamme epesentation of the BMW algeba of type A with paametes l and movethefieldq(l, ) is ieducible, except when l { 3,,, },whenitiseducible. 3 As will appea in the poof of the main theoem, the assumption that H F, (n) is semisimple is cucial. It is equivalent to the condition that k foeveyk {,...,n}: see Coollay 3.44, p. 48 of []. Some cases of educibility of the Lawence Kamme epesentation have been studied in the past, but no systematic study is done. Fo instance in [], Stephen Bigelow studies the case of educibility l = by topological methods. A consequence of ou esult and of the method that we use is the following. Theoem. Let n be an intege and let l, m and be thee nonzeo complex numbes, whee m and ae elated by m =. Suppose n 4. If k = fo some k {,...,n} o if l belongs to the set of values {, 3,,, n 3 n 3, n 3, n 3, n 3,, n 3 3 }, the BMW algeba of type A n with paametes l and m ove the field Q(l, ) is not semisimple. Suppose n = 3. If 4 = o 6 = o if l { 3,,, }, the BMW algeba of type A 3 with paametes landmovethefieldq(l, ) is not semisimple. In [4], Hans Wenzl states that the BMW algeba of type A n is semisimple except possibly if is a oot of unity o l = n, fo some n Z. Hee, Theoem gives instances in which the algeba is not semisimple. The esult of this theoem is also contained in the ecent wok of Hebing Rui and Mei Si (see [3]). They use the epesentation theoy of cellula algebas... Definitions... The BMW algeba We ecall below the defining elations of the BMW algeba B(A n ) (o simply B) oftypea n with nonzeo complex paametes l and m ove the field Q(l, ), whee is a oot of the quadatic X mx +. This algeba has two sets of (n ) elements, namely the invetible g i s that satisfy the baid elations () and () and geneate the algeba and the e i s that geneate an ideal. Fo nodes i and j with i, j n, we will wite i j if i j = and i j if i j >. The defining elations of the algeba ae as follows g i g j = g j g i if i j, () g i g j g i = g j g i g j if i j, ()

3 968 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) e i = l ( g i + mg i ) fo all i, (3) m g i e i = l e i fo all i, (4) e i g j e i = le i if i j. (5) We will also use some diect consequences of these defining elations (see [4, Poposition.]): e i g i = l e i fo all i, (6) g i = mg i + ml e i fo all i, (7) g = g i i + m me i fo all i (8) as well as the following mixed baid elations (see [4, Poposition.3]): g i g j e i = e j e i if i j, (9) g i e j e i = g j e i + m(e i e j e i ) if i j. (0)... The Lawence Kamme space We now ecall some teminology associated with oot systems of type A n. Let M = (m ij ) i j n be the Coxete matix of type A n. Let (α,...,α n ) be the canonical basis of R n and let s define a bilinea fom B M ove R n by ( ) π B M (α i,α j ) = cos. mij By the theoy in [3], B M is an inne poduct that we will simply denote by ( ). Let i denote the eflection with espect to the hypeplane Ke(α i.) of R n, and so x R n, i (x) = x (α i x)α i. Finally, let φ + denote the set of n(n ) positive oots φ + ={α,α,α + α,α 3,α 3 + α,α 3 + α + α,..., α n,α n + α n,α n + α n + +α }. We define V (n) as the vecto space ove the field F = Q(l, ) with basis the vectos x β s, indexed by the positive oots β φ +. Thus, dim F V (n) = φ + = n(n ).ThisspaceV (n) is the Lawence Kamme space (L K space).. The epesentation We define the following map on the geneatos of the BMW algeba ν (n) : B(A n ) End F ( V (n) ), g i ν i. Fo each node i, the action of ν i on x β is given as follows

4 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) x β l x β if (β α i ) = 0 (a), if (β α i ) = (b), x β+αi if (β α i ) = and (c), ν i (x β ) = x β+αi + m ht(β) x αi mx β if (β α i ) = and (c ), x β αi + m l x ht(β) α i mx β if (β α i ) = and (d), x β αi if (β α i ) = and (d ), whee (c), (c ), (d), (d ) ae the following conditions: (c) β = α t + +α i with t i, (c ) β = α i+ + +α s with s i +, (d) β = α t + +α i with t i, (d ) β = α i + +α s with s i +. We then define ν (n) (e i ) = l m (ν i + mν i id V (n)). Wehave 0 if(β α i ) = 0, ( l l )x α i if (β α i ) =, ν (n) (e i )(x β ) = x ht(β) α i if (β α i ) = and (c), ht(β) x αi if (β α i ) = and (c ), l x ht(β) α i if (β α i ) = and (d), l ht(β) x αi if (β α i ) = and (d ). We can check that the map ν (n) defines a epesentation of B(A n ) in the L K space V (n) (see [0, Chapte 7, pp ]). We notice that ν (n) (e i )(x β ) is always a multiple of x αi. This is an impotant fact to show the educibility of the epesentation fo some specializations of its paametes. 3. Reducibility of the epesentation We show that when the epesentation ν (n) is educible, the action on a pope invaiant subspace of the L K space is an Iwahoi Hecke algeba action. Indeed, we show that the e i s act tivially. We then ecall some facts about the degees of the ieducible epesentations of the Iwahoi Hecke algeba, which we assume to be semisimple. This assumption plays a key ole in the poof of the main theoem in Section 4, uling out some values fo. We will investigate whethe the Iwahoi Hecke algeba epesentations of small degees may occu in the L K space and if so fo which values of l and. We show that if thee exists a one-dimensional invaiant subspace inside V (n), it foces the value fo l, except when n = 3 when it foces l n 3 { 3, }. Convesely, we show that fo these 3 values of l and, thee exists a one-dimensional invaiant subspace of V (n) and the epesentation is thus educible. Similaly, we show that if thee exists an ieducible (n )-dimensional invaiant subspace inside V (n), it foces l = n 3 o l = in the case when n 4andl n 3 { 3,, } in the case when n = 4. Convesely, fo each of these values of l and, thee exists an ieducible (n )- dimensional subspace of V (n), which shows the educibility of the epesentation in these cases as well. We end the section by showing that when l = o l = 3, the epesentation is educible. We exhibit an invaiant subspace of the L K space that we show to be pope fo these values of l and.

5 970 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Action on a pope invaiant subspace of the L K space We show that if the epesentation is educible, then the e i s act tivially on any pope invaiant subspace of the L K space, which means the action is a Hecke algeba action. Poposition. Fo any pope invaiant subspace U of V (n),wehaveν (n) (e i )(U) = 0 fo all i. Poof. Let U be a pope invaiant subspace of V (n) and let u be a nonzeo vecto of U such that ν (n) (e i )(u) 0. Since ν (n) (e i )(u) is a multiple of x αi,weseethatx αi is in U. Fom thee, we have ν i (x αi ) = x αi +α i + mx αi mod Fx αi. Hence x αi +α i + mx αi is in U. Anothe application of ν i now yields ν i (x αi +α i + mx αi ) = x αi + m l x α i, fom which we deive that x αi is in U. By induction, we see that all the x αt s fo t i ae in U. In paticula, x α is in U. Fom thee, it is easy to see that all the x β s ae in fact in U. ThenU is the whole L K space V (n), in contadiction to ou assumption that U is pope. Coollay. Let W be a pope ieducible invaiant subspace of V (n).then,w is an ieducible H F, (n)- module. Poof. Let W be a pope ieducible invaiant subspace of V (n). By Poposition and defining elation (3), we have [ g i + mg i ].W = 0 fo all i. Hence W is an ieducible H F, (n)-module. We now ecall some geneal facts about the ieducible epesentations of the Iwahoi Hecke algeba of the symmetic goup. The following two popositions wee established by James fo the symmetic goup Sym(n). They emain tue fo the Iwahoi Hecke algeba H F, (n) since we wok in chaacteistic zeo and assume H F, (n) to be semisimple (see []). Poposition. Let n be an intege with n 7. Assume that H F, (n) is semisimple. Then, evey ieducible H F, (n)-module is eithe isomophic to one of the Specht modules S (n),s (n),s (n,),s (,n ) o has dimension geate than (n ). Poof. It follows fom Theoem 6, point (i) of [7]. We note that the statement is also tue when n = 3 and n = 5. When n = 4, the statement fails as S (,) has dimension and when n = 6, the statement also fails since S (3,3) and S (,,) both have dimension 5. When the intege n is geate than o equal to 9, thee exist even bette estimates of the dimensions of the ieducible H F, (n)-modules, as follows. Poposition 3. Let n be an intege with n 9. Assume that H F, (n) is semisimple. Then, evey ieducible H F, (n)-module is eithe isomophic to one of the Specht modules S (n),s (n,),s (n,),s (n,,) o to one of thei conjugates, o has dimension geate than (n )(n ).

6 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Poof. It follows fom Theoem 7 of [7] with N = 9. We have the coollay on the dimensions. Coollay. (i) Let n be an intege with n 3 and n / {4, 8}. Assume that H F, (n) is semisimple. Let D be an ieducible H F, (n)-module. Then, thee ae two possibilities: { dim D,n, n(n 3) (n )(n ) eithe, (n )(n ) o dim D >. (ii) Assume that H F, (4) is semisimple. Let D be an ieducible H F, (4)-module. Then dim D {,, 3}. (iii) Assume that H F, (8) is semisimple. Let D be an ieducible H F, (8)-module. Then dim D {, 7, 4, 0, } o dim D >. }, Poof. Points (ii) and (iii) can be seen diectly by using the Hook fomula. Point (i) is fo n 9a diect consequence of Poposition 3 afte noticing that S (n,) has dimension n(n 3) and S (n,,) dimension (n )(n ). Fo smalle n, the statement also holds by diect investigation using the Hook fomula. Coollay and Coollay imply that any pope ieducible invaiant subspace of the L K space V (n) n(n 3) (n )(n ) has dimension, n,, o dimension geate than (n )(n ), except in the special cases when n {4, 8}. Next, we investigate the existence of a one-dimensional invaiant subspace of V (n). We define fo two nodes i and j with i < j We will sometimes wite w i, j instead of w ij. 3.. The case l = n 3 w ij = x αi + +α j. We will show the existence of a one-dimensional invaiant subspace of the L K space when l = n 3.Wepovethefollowingtheoem. Theoem 3. Let n be an intege with n 3 and assume ( ). Suppose n 4. Thee exists a one-dimensional invaiant subspace of V (n) if and only if l = n 3. If so, it is spanned by s<t n s+t w st. Suppose n = 3. Thee exists a one-dimensional invaiant subspace of V (3) if and only if l = o l 3 = 3. Moeove, if 6, it is unique and when l = 3, it is spanned by w + w 3 + w 3, when l = 3, it is spanned by w w 3 + w 3. If 6 =, thee ae exactly two one-dimensional invaiant subspaces of V (3) and they ae espectively spanned by the vectos above.

7 97 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Poof. Let U be a one-dimensional invaiant subspace of V (n) and let u be a spanning vecto of U. Fo each i, let γ i be the scala such that ν i (u) = γ i u. Since by Poposition we have (ν +mν i i id V (n))(u) = 0, it follows that γ +mγ i i = 0. Hence γ i {, }.Futhe,since( ), the baid elation ν i ν j ν i = ν j ν i ν j when i j foces that γ i takes the same value as γ j.let sdenote by γ the common value of the γ i s. So, fo each i, wehaveν i (u) = γ u with γ {, }. A geneal fom fo u is u = i< j n μ ij w ij, whee μ ij F. We look fo elations between these coefficients. We will use the following lemma. Lemma. Let i be some node. Suppose v = k< μ f n kf w kf is a vecto of V (n) with ν i (v) = γ v, whee γ {, }. Then the following equalities hold fo the coefficients of v: s i +, μ i+,s = γμ i,s, () t i, μ t,i+ = γμ t,i. () When i =,only() holds and when i = n,only() holds. Poof. To show (), we look at the coefficient of w i+,s in ν i (v) = γ v, whee s i +. We get μ i,s mμ i+,s = γμ i+,s.sinceγ + m = γ, this equality is equivalent to μ i+,s = γμ i,s. Similaly, by equating the coefficients of w t,i+ in ν i (v) = γ v, we obtain (). Applying these equalities to the coefficients of u, we see that all the coefficients of u must be nonzeo. In paticula, when n 4, the coefficient μ 34 of u is nonzeo. Because an action of g on w 34 is a multiplication by and an action of g on the othe tems w ij does not ceate any tem in w 34, this foces γ =. Thus, without loss of geneality, we have u = i+ j w ij. i< j n Fom thee, we look at the action of g on u and the esulting coefficient in w. The action of g on w is a multiplication by l and an action of g on the w, j s fo 3 j n ceates new tems in w with espective coefficients m j 3. Thus, we get the equation: 3 l + m n ( ) j = 4, j=3 fom which we deive that l =. n 3 Convesely, if l =,wedefineu as n 3 i< j n i+ j w ij and check that ν i (u) = u fo each i. Fo details, see [0, 8.]. This ends the poof of the theoem when n 4. The case n = 3 is diffeent in that γ can take eithe values o focing in one case l = and in the othe case l 3 = 3. Details appea in [0, 8.].

8 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) The case l { n 3, n 3 } In Theoem 4, the case n = 3 was special. Likewise, in the following theoem, the case n = 4needs to be fomulated sepaately. Theoem 4. Let n be a positive intege with n 3 and n 4. Assume H F, (n) is semisimple. Then, thee exists an ieducible (n )-dimensional invaiant subspace of V (n) if and only if l = n 3 o l =. n 3 If so, it is spanned by the v (n) s, i n,wheev (n) is defined by the fomula: i i ( v (n) = i ) w i,i+ + l n (w s i i,s ) w i+,s s=i+ i + ɛ l (w n i +t t,i ) w t,i+ t= with ɛ n 3 =, ɛ n 3 =. Suppose n = 4 and assume H F, (4) is semisimple. Then, thee exists an ieducible 3-dimensional invaiant subspace of V (4) if and only if l {,, 3 }. If l {, }, it is spanned by v(4),v(4),v(4) 3. If l = 3, it is spanned by the vectos: ( u = w 3 + w ) 3 w 34 w 4 w 4, u = w w 3 w 34 ( w ) w 4, u 3 = ( + 3) w + w 3 w 3 + w 4 w 4. Poof. Suppose that thee exists an ieducible (n )-dimensional invaiant subspace U of V (n). Claim. Except in the case when n = 6, thee exists a basis (v,...,v n ) of U such that one of the following two sets of elations holds ( ) ( ) ν t (v i ) = v i, t / {i, i, i + }, ν i (v i ) = v i, i n, ν i+ (v i ) = (v i + v i+ ), i n, ν i (v i ) = v i + v i, i n, ν t (v i ) = /v i, t / {i, i, i + }, ν i (v i ) = v i, i n, ν i+ (v i ) = /(v i + v i+ ), i n, ν i (v i ) = /v i v i, i n.

9 974 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Poof. By Poposition, thee ae exactly two inequivalent ieducible epesentations of H F, (n) of degee (n ), exceptinthecasen = 6, when thee ae exactly fou inequivalent ieducible epesentations of H F, (6) of degee 5. Conside now the set of elations ( ) (esp. ( )). Fo each i, letm i (esp. N i ) be the matix of the endomophism ν i in the basis (v,...,v n ). It is a staightfowad veification that the M i s (esp. N i s) satisfy the baid elations and the elation M + mm i i = I n (esp. N + mn i i = I n )foeachi, whee I n is the identity matix of size (n ). HencetheM i s (esp. the N i s) yield a matix epesentation of H F, (n) of degee (n ). To show that these two matix epesentations ae ieducible, elying on Poposition, it suffices to check that thee is no one-dimensional invaiant subspace of F n. This is the case when n. When n = 3, the two matix epesentations ae equivalent. When n 4, they ae not: visibly, the matices of one epesentation all have the same tace (n ) + and the matices of the othe epesentation all have thesametace(n ). These two values ae distinct when ( ) and n 4. We conclude that these two matix epesentations ae the two inequivalent ieducible epesentations of H F, (n) when n 4 and n 6. Fo n 4, we can show that it is impossible to have the second set of elations, except in the case n = 4 when it foces l = 3. Fo a detailed poof of this fact, see [0, Chapte 8, pp. 8 95]. Let n 3 and suppose that the v i s satisfy ( ). Theelationν i (v i ) = v i implies that in v i thee ae no tems in w ts fo s i ot i + ot i and s i +. Thus, a geneal fom fo v i must be v i = μ i,i+ w i,i+ + n s=i+ μ i,s w i,s + n s=i+ μ i+,s w i+,s i i + μ t,i w t,i + μ t,i+ w t,i+. (3) t= t= Since ν i (v i ) = v i, both equalities () and () hold with γ =.Futhe,sinceν q(v i ) = v i fo q / {i, i, i + }, applying () and () with i = q and γ = yields Apply (4) with q i and j {i, i + } to get j q +, μ q+, j =,μ q, j, (4) k q, μ k,q+ = μ k,q. (5) q i, μ q+,i = μ q,i & μ q+,i+ = μ q,i+. Apply (5) with q i + and k {i, i + } to get q i +, μ i,q+ = μ i,q & μ i+,q+ = μ i+,q. Expession (3) can now be ewitten as follows. v i = ζ (i) w i,i+ + δ (i) n (w s i i,s ) i w i+,s + λ (i) (w t t,i ) w t,i+, s=i+ whee ζ (i), δ (i) and λ (i) ae thee coefficients to detemine. Fist, we show that all the δ (i) with i {,...,n } may be set to the value one. Notice that if v,...,v n satisfy ( ), thenδv,...,δv n also satisfy ( ), whee δ is any nonzeo scala. Then, without loss of geneality, we set δ () =. t=

10 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Suppose δ (i) = fo some node i with i n. We will show that δ (i+) =. Notice that δ (i+) is the coefficient of w i+,i+3 in v i+. Since an action of g i+ on v i neve ceates a tem in w i+,i+3, by looking at the coefficient of w i+,i+3 in ν i+ (v i ) = v i + v i+,weget0= δ (i) + δ (i+).afte eplacing δ (i) by, this yields δ (i+) =. Thus, all the δ (i) may be set to the value. It emains to find the coefficients ζ (i) and λ (i). By looking at the coefficient of w i,i+ in ν i+ (v i ) = (v i + v i+ ),we get ζ (i) + i λ (i+) =, fo each i with i n. (6) Also, by looking at the coefficient of the same tem w i,i+ in the elation ν i (v i ) = v i + v i, we get mζ (i) i 3 λ (i) = ζ (i), fo each i with i n. Afte multiplication by a facto,weobtain ζ (i) + i λ (i) =, fo each i with i n. (7) By (6) and (7), we get λ (i) = i λ (),foalli. Futhe, change indices in (6) to get ζ (i ) + i λ (i) = fo each i with i n. (8) Now (7) and (8) show that ζ (i) = ζ (i ) fo each i with i n. In othe wods, all the ζ (i) ae equal with a cetain scala ζ. The elation between ζ and λ () is given by Eq. (8) with i = λ () = ζ. (9) Thus, by detemining ζ, we will get a complete expession fo all the vectos v i s. Since we have v = ζ w + n (w s 3,s ) w,s, s=3 by looking at the coefficient of w in the elation ν (v ) = v,wegettheequation: ( ζ l + ) = ( ) n 3. (0) Futhe, by looking at the coefficient of w i,i+ in ν i (v i ) = v i,wehave ( ζ l + ) = n i s i 3 m s i + λ (i) s=i+ m t l i t t= i.e. ( ζ l + ) = ( ) n i + λ (i) l ( i i ). ( ) i

11 976 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Now wite down ( ) and ( ) 3 : ( ζ l + ( ζ l + ) = ( ) n 4 + λ () ) = ( ) n 5 + λ () l l ( ), ( ) ( ) 3, ( ) 3 whee λ (3) has been eplaced with λ(). Subtact these two equalities to get λ () l ( 4 ) = ( ) ) n 4 (. ( ) ( ) 3 Afte multiplying this equality by and dividing it by 4 (ecall that m 0), we obtain λ () = l ( ) n 3. Hence, by (9), ζ = l( ) n 3. Plugging this value fo ζ into (0) now yields l = {, ( hence l ) n 3, }. n 3 n 3 If l =, we get successively n 3 λ() = n 3 = l, ζ = and λ (i) = n i. l If l =, n 3 λ() and ζ ae still espectively and l and λ (i) = n i. l We obtain the fomula announced in Theoem 4. Convesely, if l {, }, we can show that the v (n) s defined in Theoem 4 satisfy the elations ( ) (see [0, 8.3]). In paticula, thei linea span ove F is a pope invaiant subspace of V (n), n 3 n 3 i hence is an H F, (n)-module by Coollay. When n 4, if the vectos v (n) s wee linealy dependent, then thei linea span would eithe be one-dimensional o would contain a one-dimensional i H F, (n)-submodule, as thee is no ieducible H F, (n)-module of dimension between and (n ) by Coollay. In any case, by Theoem 3, that would foce l = when n 3 and l n 3 { 3, } 3 when n = 3. This is impossible with ou assumption that l {, } and the fact that n. As n 3 n 3 fo n = 4, the feedom ove F of the family of vectos (v (4), v(4), v(4) 3 ) is a staightfowad veification. We ae now able to conclude: the vecto space Span F (v (n),...,v(n) n ) is (n )-dimensional, is invaiant unde the action by the g i s and is an H F, (n)-module since it is a pope invaiant subspace of V (n). Then, by the elations satisfied by the v (n) s, it must be ieducible. i To complete the poof of Theoem 4, we show that thee does not exist any ieducible 5- dimensional invaiant subspace of V (6) that is isomophic to one of the Specht modules S (3,3) o S (,,). Indeed, suppose such a subspace exists and name it W. Since we have assumed that H F, (6) is semisimple, we may use the banching ule as it appeas in Coollay 6. of []. We have S (3,3) HF, (5) S (3,), S (,,) HF, (5) S (,,). We will show that the estiction of W to H F, (5) cannot be isomophic to S (3,) o S (,,),hence a contadiction. A poof of the following fact is in [0, 8.3].

12 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Fact. Suppose H F, (5) is semisimple. Then, up to equivalence, the two ieducible matix epesentations of degee 5 of H F, (5) ae espectively defined by the matices P, P, P 3, P 4 and Q, Q, Q 3, Q 4 given by P :=, P :=, P 3 :=, P 4 :=, whee the blanks must be filled with zeos; the matices Q i s ae defined fom the matices P i s by eplacing with. In [0, Chapte 8, pp. 5 8], it is shown that it is impossible to have a basis (w, w, w 3, w 4, w 5 ) of W in which the matices of the left action by the g i s, i =,...,4aetheQ i s. In paticula, by consideing W as a subspace of V (5) insteadofasubspaceofv (6), the computations of [0] also show that: Result. The ieducible matix epesentation of degee 5 of H F, (5) defined by the matices Q i s is not a constituent of the Lawence Kamme epesentation of degee 0 of the BMW algeba of type A 4. Suppose now that thee exists a basis (w, w, w 3, w 4, w 5 ) of W in which the matices of the left action by the g i s, i =,...,4aetheP i s. We ead fom the matices P and P 3 that g.w 4 = w 4 and g 3.w 4 = w 4. Thus, we have w 4 = μ (4) 3 w 3 + μ (4) 3 w 3 + μ (4) 4 w 4 + μ (4) 4 w 4, whee the coefficients ae elated by μ (4) 4 = μ(4) 3 = μ(4) 3 = μ(4) 4. In paticula, all these coefficients ae nonzeo. The othe spanning vectos of W ae w 5 = g 4.w 4, w = g.w 4 w 4, w = g 4.w, w 3 = g 3.w w. A quick glance at these equations shows that the node numbe 6 neve appeas in any of the w i s. But W is an invaiant subspace of V (6). In paticula, it must be invaiant unde the action by g 5.This is not compatible with the expession fo w 5. We conclude that it is impossible to have W HF, (5) S (3,) o W HF, (5) S (,,)

13 978 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) and so W cannot be isomophic to S (3,3) o S (,,). Thus, by the fist pat of the poof, the existence of an ieducible 5-dimensional invaiant subspace of V (6) implies that l {, }. This completes 3 3 the poof of the theoem. As in Result, by consideing W as a subspace of V (5) insteadofasubspaceofv (6),wecanshow the following esult. Result. Assume H F, (5) is semisimple. If thee exists an ieducible 5-dimensional invaiant subspace of V (5) then l = The cases l = andl= 3 In this section, we show that when l = the epesentation ν (n) is educible fo all n 4 and when l = 3, the epesentation is educible fo all n 3. We intoduce an invaiant subspace of V (n) that we show to be nontivial fo these values of l and. Poposition 4. Fo any two nodes i and j with i < j n, define { c ij = g j...g i+ e i g i+...g j if j i +, c i,i+ = e i. Then, K (n) = i< j n Ke ν(n) (c ij ) is an invaiant subspace of V (n). Moeove, any pope invaiant subspace of V (n) must be contained in K (n). Poof. K (n) is not the whole L K space, as is visible fom the expessions fo ν (n) (e i ).Wecancheck that K (n) is a B-module. Veification of this fact is tedious and can be found in [0, ]. Let W be a pope invaiant subspace of V (n).bypoposition,wehaveν (n) (c i,i+ )(W) = 0foalli with i n. Then ν (n) (c i, j )(W) = 0foalli and j with i < j n. HenceW must be contained in K (n). To show that ν (n) is educible, it will suffice to exhibit a nontivial element in K (n) when l = o l = 3. The following poposition shows that K (4) is ieducible when l = and H F, (4) is semisimple. Poposition 5. Assume H F, (4) is semisimple. Thee exists an ieducible -dimensional invaiant subspace of V (4) if and only if l =. If so it is unique and it is K (4). Moeove, it is spanned ove F by the two linealy independent vectos: v = w 3 w 3 + w 4 w 4, v = w w 3 w 4 + w 34. Poof. See the poofs of Result, p. 54 and Coollay 5, p. 59 of [0]. The next poposition shows the educibility of the epesentation when l = and n 4. Poposition 6. Assume l =. Then the vecto v = w 3 w 3 + w 4 w 4 of Poposition 5 belongs to K (n) fo all n 4.Thus,ν (n) is educible fo evey n 4.

14 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) Poof. Fo n = 4, the esult is contained in Poposition 5. When i 5, we simply have fo any j i +, ν i+...ν j (v ) = v j i and since ν (n) (e i )(v ) = 0, we see that v is thus annihilated by all the ν (n) (c ij ) s with i 5. Also, as we saw in Poposition 5, the vecto v is in K (4), hence it is annihilated by all the ν (n) (c ij ) s with j 4. Thus, it suffices to check that v is annihilated by ν (n) (c j ), ν (n) (c j ), ν (n) (c 3 j ) and ν (n) (c 4 j ) fo all j 5. We will use the following fomulas that give the action of the c ij s on the basis vectos of the L K space in some elevant cases hee. ν (n) (c ij )(w i, j k ) = l k w ij, (R k ) k j i, ν (n) (c ij )(w i k,i ) = (k )+( j i ) w ij, (L j i,k ) k i, ( ν (n) (c ij )(w i t, j s ) = )( t+s t+s 3 l ) w ij, (C t,s ) t i, s j i. These fomulas can be obtained by using the isomophism between the BMW algeba and the tangle algeba of Moton and Taczyk (see []). The use of the tangles allows us to deive algebaic elations by a geometic appoach as in Appendix C of [0]. When l =, wenotethattheactionofc i, j on w i t, j s is zeo. Fom thee, we have fo j 5, whee we eplaced l by : ( ν (n) (c, j )(v ) = ν (n) (c, j ) w 3 ) w 4 = 0 by(r j 3 ) and (R j 4 ), ( ν (n) (c, j )(v ) = ν (n) (c, j ) w 3 + ) w 4 = 0 by(r j 3 ) and (R j 4 ), ( ν (n) (c 3, j )(v ) = ν (n) (c 3, j ) w 3 ) w 3 = 0 by(l j 3, ) and (L j 3, ), ( ν (n) (c 4, j )(v ) = ν (n) (c 4, j ) w 4 ) w 4 = 0 by(l j 4, ) and (L j 4,3 ). So v is in K (n) fo all n 4, as announced. It will be useful to notice that by the game of the coefficients, the equalities to the ight of the fist two lines of equations still hold when l = 3. When l = 3, we have a simila esult. Poposition 7. When l = 3, the vecto u defined as in Theoem 4 by the expession u = w 3 + w 3 + ( + )w 3 34 w 4 w 4 belongs to K (n) fo all n 4. Thus,whenl= 3, the epesentation ν (n) is educible fo evey n 3. Poof. When l = 3, ν (3) is educible by Theoem 3 and ν (4) is also educible by Theoem 4. Suppose now n 5. To show that u is in K (n), like in the case l =, it will suffice to check that ν (n) (c ij )(u ) = 0foalli 4 and j 5. With l = 3, the coefficients of type (C t,s ) ae no longe zeo. But we have: ν (n) (c, j )(w 3 w 4) = 0by(C, j 3 ) and (C, j 4 ).Foν (n) (c 3, j )(u ), thee is no shotcut and a complete evaluation must be pefomed. We have, whee we espected the same ode of the tems in Poposition 7 fo the coefficients: [ ν (n) (c 3, j )(u ) = + j 4 + ( + j 3 ( j 3 j 5 )( j 5 )( + )] 3 w 3, j. ) ( + )( j 4 j 6 + ) 3

15 980 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) by ules (L j 4,), (L j 3 j 4, ) and (L j 4,3 ) The ules used ae, in the same ode: (L j 3, ), (L j 3, ), (R j 4 ), (C, j 4 ) and (C, j 4 ). All the coefficients cancel nicely to give ν (n) (c 3, j )(u ) = 0. Finally, fo ν (n) (c 4, j )(u ), only the tems in u whose last node is node numbe 4 yield a nonzeo contibution, the fist one contibuting with a coefficient ( + ) 3 j 5, the second one with a coefficient and the thid one with a coefficient j 4 espectively. The sum of these thee coefficients is zeo. Thus, we ae done with all the cases and conclude that u belongs to K (n) fo all n 4. At this stage, we have shown that when l and take the values of Theoem, the epesentation ν (n) is educible. In the next pat, we show convesely that if ν (n) is educible, then l and must be elated in the way descibed in Theoem. 4. Poof of the main theoem We pove the main theoem on the epesentation ν (n). We then show that ν (n) is equivalent to the Lawence Kamme epesentation of the BMW algeba. The idea is to pove the main theoem sepaately fo small values of n and to use induction fo lage n. Let s assume that the main theoem is tue fo n {3, 4, 5, 6}. These cases will be dealt with sepaately. Given an intege n with n 7, suppose that the main theoem holds fo ν (n ) and fo ν (n ). We saw in pat 3 that when l {, 3,,, }, the epesentation ν (n) is educible. We will show convesely that if ν (n) n 3 n 3 n 3 is educible, it foces these values fo l and. Suppose ν (n) is educible and let W be an ieducible invaiant subspace of V (n). By Coollay, W is an ieducible H F, (n)-module. Suppose fist n = 7on 9. So dim W {,n, n(n 3), (n )(n ) } o dim W > (n )(n ) (see Coollay ). If dim W =, Theoem 3 implies that l = n 3.Also,ifdimW = n, Theoem 4 implies that l {, }. Suppose now that l / {,, }.ThenwehavedimW n(n 3).We n n n 3 n 3 n 3 show that this bound is lage enough to make the intesection of W with the L K spaces V (n ) and V (n ) nontivial. Indeed, we have the following esult. Claim. Let W be a subspace of V (n). If dim W > n,thenw V (n ) {0}. If dim W > n 3,thenW V (n ) {0}. Poof. If W V (n ) ={0}, thel KspaceV (n) contains the diect sum W V (n ), which yields on the dimensions: dim W + (n )(n ) n(n ).ThendimW n. Similaly, if W V (n ) ={0}, we get dim W n(n ) (n )(n 3) = n 3. Lemma. When n > 6,wehave n(n 3) > n 3 and n(n 3) > n. By the claim and the lemma, the intesections W V (n ) and W V (n ) ae both nontivial. Since W is not the whole space V (n), it cannot contain V (n ) by the aguments of the poof of Poposition. Similaly, it cannot contain V (n ).HenceW V (n ) (esp. W V (n ) )isapope invaiant subspace of V (n ) (esp. V (n ) ). This implies that ν (n ) and ν (n ) ae both educible. Since we assumed the main theoem to be tue fo ν (n ) and ν (n ),weget { l, 3,, n 5, n 4 n 4 } {, 3,, n 7, n 5 n 5 }. Since, (n 3) and n when H F, (n) is semisimple, this only leaves the possibility l {, 3 }.

16 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) It emains to deal with the case n = 8. Suppose n = 8. If l / {,, },thenwehavedimw > 3 = 8 3. Hence, the same aguments as befoe apply and yield again l {, 3 }. Thus, we have shown that if ν (n) is educible and if l and ae such that l / {,, }, n 3 n 3 n 3 then l {, 3 }.Sowehavepoventhatifν (n) is educible, then l {, 3,,, }. n 3 n 3 n 3 We now show that the main theoem holds fo ν (n) whee n {3, 4, 5, 6}. Thecasen = 3 follows fom Theoem 3 and Theoem 4 and the case n = 4 fom Theoem 3, Poposition 5 and Theoem 4. Fo the case n = 5, we efe the eade to [0,, pp. 6]. As fo the case n = 6, it must be slightly adapted fom the geneal case. Indeed, suppose ν (6) is educible and let W be an ieducible invaiant subspace of V (6) with dim(w) 9. If dim(w)>9, then Claim implies that W V (4) {0}. If dim(w) = 9, Claim does not apply, but we notice that dim(w) + dim(v (4) ) = dim(v (6) ).Thus, if W V (4) ={0}, we then getw V (4) = V (6). By Poposition, we haveν (6) (e 5 )(W) = 0. But e 5 also acts tivially on V (4),henceactstiviallyonV (6). This is a contadiction. So again we have W V (4) {0}, and the est of the poof is the same as in the geneal case. To end the poof of the main theoem, we show that ν (n) is equivalent to the Lawence Kamme epesentation of the BMW algeba. Above, we poved that ν (n) is educible if and only if l {, 3,,, }.Inpaticula,ν (n) is geneically ieducible ove Q(l, ). Futhe,we n 3 n 3 n 3 notice that ν (n) factos though the quotient B/I whee I is the two-sided ideal geneated by all the poducts e i e j with i j >. Also, if I denotes the two-sided ideal of B geneated by e,we obseve that I is not in the kenel of ν (n). Then, ν (n) is an ieducible epesentation of I /I of degee n(n ). To conclude, we use the wok of Cohen, Gijsbes and Wales. In [4], they show that thee ae only two inequivalent ieducible epesentations of I /I of degee n(n ).Oneofthemis the Lawence Kamme epesentation of the BMW algeba. The epesentation ν (n) is equivalent to that epesentation. Ou is the of [4]. 5. Non-semisimplicity of the BMW algeba fo some specializations of its paametes Replacing the L K epesentation by one in which is eplaced by its algebaic conjugate gives anothe epesentation. We call it the conjugate L K epesentation. By the symmety of the oles played by and, when n 4 and H F,(n) is semisimple, the conjugate L K epesentation is educible if and only if l {,, n 3, n 3, n 3 }. In paticula, fo n 6, since 3 / {, 3,,, }, the two epesentations ae not equivalent. This is also tue when 3 n 3 n 3 n 3 n {4, 5}. Fo instance, fo the L K epesentation, the tace of the matix of the left action by g n is (n )(n 3) + (n )m (see fo instance [0, Chapte 6, p. 55]). Fo the conjugate epesentation l it is (n )(n 3) ( ) + (n )m. l We note that Poposition emains valid fo the conjugate L K epesentation. A consequence of this poposition is that when the epesentation is educible, it is indecomposable. Then the BMW algeba is not semisimple fo the values of l and fo which the L K epesentation o its conjugate epesentation ae educible. Finally, the Iwahoi Hecke algeba H F, (n) is a quotient of the BMW algeba B(A n ).Thus,ifH F, (n) is not semisimple, B(A n ) is not semisimple eithe. Theoem is thus poven. Refeences [] S. Bigelow, Baid goups ae linea, J. Ame. Math. Soc. 4 (00) [] S. Bigelow, The Lawence Kamme epesentation, axiv:math/004057v, 00. [3] N. Boubaki, Goupes et algebes de Lie, Chapites 4, 5 et 6, Masson, 98. [4] A.M. Cohen, D.A.H. Gijsbes, D.B. Wales, BMW algebas of simply laced type, J. Algeba 85 () (005) [5] A.M. Cohen, D.B. Wales, Lineaity of Atin goups of finite type, Isael J. Math. 3 (00) 0 3. [6] F. Digne, On the lineaity of Atin baid goups, J. Algeba 68 () (003) [7] G.D. James, On the minimal dimensions of ieducible epesentations of symmetic goups, Math. Poc. Cambidge Philos. Soc. 94 (983) [8] D. Kamme, Baid goups ae linea, Ann. of Math. 55 (00) [9] R. Lawence, Homological epesentations of the Hecke algeba, Comm. Math. Phys. 35 (990) 4 9.

17 98 C. Levaillant, D. Wales / Jounal of Algeba 33 (00) [0] C. Levaillant, Ieducibility of the Lawence Kamme epesentation of the BMW algeba of type A n, Califonia Institute of Technology, PhD dissetation, 008, [] A. Mathas, Iwahoi Hecke Algebas and Schu Algebas of the Symmetic Goup, Univ. Lectue Se., vol. 5, Ame. Math. Soc., 999. [] H.R. Moton, A.J. Wassemann, A basis fo the Biman Wenzl algeba, pepint, 989. [3] H. Rui, M. Si, Gam deteminants and semisimple citeia fo Biman Muakami Wenzl algebas, J. Reine Angew. Math. 63 (009) [4] H. Wenzl, Quantum goups and subfactos of type B, C, and D, Comm. Math. Phys. 33 (990)