Representation Theory of the Temperley-Lieb Algebra and its connections with the Hecke Algebra

Transcription

1 Representation Theory of the Temperley-Lieb Algebra and its connections with the Hecke Algebra Tim Weelinck July 19, 2012 Bachelor Thesis Supervisor: dr. Jasper V. Stokman KdV Instituut voor Wiskunde Faculteit der Natuurwetenschappen, Wiskunde en Informatica Universiteit van Amsterdam

2 Abstract In this thesis we will study the representation theory of the Temperley-Lieb algebra T L n (q). We begin by introducing the Hecke algebra H n (q) as a deformation of the group algebra C[S n ] of the symmetric group S n. The Hecke algebra is dependent on a parameter q C and for so-called generic q the Hecke algebra is semisimple and isomorphic to the group algebra. We are able to study the representation theory of the Hecke algebra in a similar manner to that of the symmetric group and find irreducible representations of H n (q) labeled by the partitions λ n analogous to the Young orthogonal representations. We then introduce the Temperley-Lieb algebra, also dependent on a parameter q C and prove that T L n (q 1 2 ) is isomorphic to a quotient of the Hecke algebra. By inquiring which representations factorize through the quotient we find that the H n (q) representations labeled by a Young diagram with two rows or less constitute a complete set of irreducible representations of T L n (q 1 2 ). Finally we try to connect these irreducible representations to the T L 2n (q 1 2 )-module built from the perfect non-crossing matchings on [2n], denoted V F2n. We formulate, and give evidence for the conjecture that V F2n is irreducible and isomorphic to the irreducible representation of T L 2n (q 1 2 ) labeled by the partition λ = (n, n) 2n. Data Title: Representation Theory of the Temperley-Lieb Algebra and its connections with the Hecke Algebra Author: Tim Weelinck, timweelinck@gmail.com, Supervisor: dr. Jasper V. Stokman Second assessor: dr. Hessel Posthuma Date: July 19, 2012 Korteweg de Vries Instituut voor Wiskunde Universiteit van Amsterdam Science Park 904, 1098 XH Amsterdam

3 Contents 1 A Visual Introduction 2 2 The Hecke Algebra Coxeter systems Preliminaries on associative algebras, modules and semisimplicity Construction of the Hecke algebra Semisimplicity of the Hecke algebra for generic q Representation theory of the Hecke algebra Representation theory preliminaries Partitions, Young diagrams and tableaux Young orthogonal representations of Hecke algebras of type A n The Temperley-Lieb Algebra The Temperley-Lieb algebra as an abstract algebra The Temperly-Lieb algebra as a quotient of the Hecke algebra Representation theory of the Temperley-Lieb algebra The action on matchings as a module and its resolution in irreducibles Populaire Samenvatting 59 Bibliography 64 1

4 Chapter 1 A Visual Introduction Almost all mathematical structures, if not all, mentioned in this thesis are somehow interrelated to Emil Artin s braid group on n strands, denoted B n. The group in some sense generalizes the symmetric group on n symbols, denoted S n. Consider two sets of n points, each arranged in a horizontal line, and connect each point to a point from the other set in a one-to-one correspondence. We have therefore n connecting lines, or strands, connecting every point to a unique neighbor point in the other set. Such a connection is called a braid. For n = 1 there is only one braid possible However for n = 2 we already have infinitely many different braids. This is so because it also matters how the points are connected as well as which point is connected to which. The following two braids are different for instance (1.1) Any two braids can be composed by drawing the first braid below the second, identifying the points in the middle and connecting the strands. If we were to compose the two braids above = 2

5 as we are allowed to straighten out strands. The above composition of braids is a group operation and the group of braids on n strands is denoted B n. For example the identity braid is given by the braid with only vertical strands and the two braids in 1.1 are each others inverse. Consider B 4 where you can have complicated looking braids such as the following one Nevertheless all braids can be built from simple intertwinings of strands. For B 4 all braids can be built from the following three braids (simple intertwinings) and their inverses These simple intertwiners are the generators of B 4 and we denote them respectively σ 1, σ 2, σ 3. In general B n has generators σ 1,..., σ n 1 and they have the following defining relations For instance for σ 1 σ 3 = σ 3 σ 1 in B 4. σ i σ j = σ j σ i if i j > 1 (1.2) σ i σ i±1 σ i = σ i±1 σ i σ i±1 (1.3) = Of course (1.2) and (1.3) are reminiscent of the defining relations of the generators of the symmetric group s 2 i = 1, s i s j = s j s i if i j > 1 s i s i±1 s i = s i±1 s i s i±1 3

6 However σ 2 i 1 for the braid group as the strands get more entangled instead of less. If we consider how the group algebra C[S n ] of the symmetric group is built from the group algebra C[B n ] of the braid group we find that we need to take a quotient to enforce this last identity C[S n ] = C[B n ]/ σ 2 i 1 Next we will introduce the Hecke algebra H n (q), named after german mathematician Erich Hecke. An associative algebra over C with unit generated by elements T 1,..., T n 1 with defining relations T i T i+1 T i = T i+1 T i T i+1 T i T j = T j T i if i j > 1 T 2 i = (q 1)T i + q where q C. We will introduce the Hecke algebra in chapter 2 as a deformation of the group algebra of the symmetric group and we will find an isomorphism between C[S n ] and H n (q) for so-called generic q. However the Hecke algebra could also have been introduced as an independent quotient of C[B n ] with its own right raison d être. H n (q) = B n / σ 2 i (q 1)σ i q At last we will describe the Temperley-Lieb algebra T L n (q), named after American mathematical physicists Neville Temperley and Elliott Lieb. An associative algebra over C with unit and generators e 1,..., e n 1 with defining relations e 2 j = (q + q 1 )e j, e j e j±1 e j = e j, e j e i = e i e j when i j > 1 where q C. This algebra has a nice visual interpretation which is somewhat alike the one of the braid group. Here the generators e 1, e 2 of T L 3 (q) are visualized as follows e 1 e 2 where the multiplication again is given by placing one diagram beneath the other. For instance the identity e 1 e 2 e 1 = e 1 = 4

7 Although you might not see it directly, the Temperley-Lieb algebra is a quotient of the braid group as well, as we will prove that and therefore T L n (q 1 2 ) = Hn (q)/ T 1 T 2 T 1 + T 1 T 2 + T 2 T 1 + T 1 + T T L n (q 1 2 ) = Bn / σ 2 i (q 1)σ i q, σ 1 σ 2 σ 1 + σ 1 σ 2 + σ 2 σ 1 + σ 1 + σ Our starting point will be studying the Hecke algebra which arises in the context of these beautiful braids. Throughout developing the representation theory it may seem we have lost this visual interpretation by taking a quotient of the braid group and traded it in for seemingly endless calculations. However I would like to reassure the reader who might grow weary of these computations that beautiful diagrams will resurface in the final chapter and will even be connected with yet another visual aid that will be introduced along the way called the Young diagram. 5

8 Chapter 2 The Hecke Algebra Before discussing the Temperley-Lieb Algebra we shall introduce the Hecke algebra: an algebra depending on a parameter q C as a deformation of a group algebra. We shall see that for so called generic q the group algebra and corresponding Hecke algebra are isomorphic. After proving its semisimplicity for these generic q we shall construct concrete representations of the Hecke algebra of the symmetric group completely analogous to the Young orthogonal representations of the symmetric group. 2.1 Coxeter systems In the construction of the Hecke algebra, denoted H(q), we follow the same route as Humphreys[1] where the Hecke algebra is defined for arbritrary Coxeter systems. Definition 2.1. A Coxeter system is a pair (W, S) consisting of a group W and a finite set of generators S W subject only to relations of the form (ss ) m(s,s ) = 1, where m(s, s) = 1 and m(s, s ) = m(s, s) 2 for s s s, s S or m(s, s ) = when there is no relation between s and s. W is called the Coxeter group, with rank S. We shall thus only look at Coxeter groups with finite rank i.e. where W has but finitely many generators. The symmetric group, S n, is a well know Coxeter group. The symmetric group S n on a finite set of n symbols is the group whose elements are permutations of the n symbols, bijections from the set {1,..., n} to itself, where the 6

9 group operation is given by the composition of two permutations. As there are n! permutations of n symbols, S n has order n! i.e. S n has n! elements. An element σ S n is called a cycle when there are k different numbers a 1,...a k such that σ(a 1 ) = a 2, σ(a 2 ) = a 3,..., σ(a k ) = a 1 and σ(b) = b for b {1,..., n} \ {a 1,..., a k }. We write σ = (a 1 a 2...a k ). Proposition 2.2. Any element s S n can be written as a product of disjoint cycles. The product is unique up to order of cycles. For S n define s 1,..., s n 1 S n where s i = (i, i + 1). Thus s i is the permutation which swaps the neighbors i and i + 1. Any cycle σ = (a 1...a k ) can be written as a product of these permutations since σ = (a 1 a 2 )(a 2 a 3 )...(a k 1 a k ) and hence s 1,..., s n 1 generate S n. Definition 2.3. We then are able to define the sign, ε(σ), of a permutation σ. The sign map ε : S n {1, 1} is the unique group homomorphism defined by ε(s i ) = 1 for 1 i n 1. The generators s 1,...s n 1 have relations s 2 i = 1, s i s j = s j s i if i j > 1 s i s i+1 s i = s i+1 s i s i+1 With these relations we find corresponding Coxeter relations (s i s i ) 1 = 1, (s i s j ) 2 = 1 if i j > 1 (s i s i+1 ) 3 = 1 Write S = {s 1,..., s n 1 }. We have found that S n is the quotient of a Coxeter group W. However we can easily see that these new relations are equivalent with the old ones and hence that the symmetric group is a Coxeter system itself. Since s 2 i = 1 is an old relation as well as a new one, we will use it to 7

10 show that these new relations induce the old ones. s i s j = s i s j 1 = s i s j (s j s i ) 2 = s i s j s j s i s j s i = s j s i when i j > 1 s i s i+1 s i = s i s i+1 s i 1 = s i s i+1 s i (s i s i+1 ) 3 = s i s i+1 s i s i s i+1 s i s i+1 s i s i+1 = s i+1 s i s i+1 The old and new relations are equivalent and therefore (S n, S) is indeed a Coxeter system. As generators s i S have order 2 in W in general each w 1 in W can be written as w = s i1 s i2 s ir for some s ij (not necessarily different) in S. We can then define the length l(w) of w, such that l(w) equals the smallest r for which w = s i1 s i2 s ir for some s ij. By convention l(1) = 0. Call any expression of w as a product of l(w) generators a reduced expression. Note that in general reduced expressions need not be unique. 2.2 Preliminaries on associative algebras, modules and semisimplicity In this section we shall introduce the theory needed to understand what is meant by an associative algebra over C with unit, left or right modules of such algebras and when an algebra is called semisimple. Definition 2.4. An associative algebra over C with unit element 1 is a vector space A over C endowed with a C-bilinear multiplication map A A A, (a, a ) a a (i.e, for all a A the maps A A given by a a a and a a a are C-linear), such that 1 a = a 1 = a and satisfying associativity a (a a ) = (a a ) a a, a, a A 8

11 Example 2.5. (i) Define for a complex vector space V, End C (V ) := {ϕ : V V ϕ linear}. Then End C (V ) is a C-algebra with multiplication the composition of linear endomorphisms. (ii) The group algebra C[G] of a finite group G is a C-algebra with multiplication ( ) µ g e g = λ g µ g e gg where λ g, µ g C. g G λ g e g, g G g,g G Definition 2.6. Let A be an associative C-algebra, a left module over A is a vector space V over C together with a bilinear map A V V, (a, v) a v such that a, a A, v V, a (a v) = (aa ) v, 1 v = v A similar definition can be given for right actions; defining a right module. Example 2.7. (i) Let V be a complex vector space. Together with the map End C (V ) V V, (ϕ, v) ϕ(v) V becomes a left End C (V )-module. (ii) Let A be an associative C-algebra. A considered as a vector space becomes a left, and right, A-module with A acting on itself by multiplication in A. This is called its regular representation and the bimodule is denoted A reg. Just as a group can sometimes be described by generators and their relations an algebra can sometimes be defined by specifying its generators and corresponding relations. Example 2.8. The group algebra C[S n ] is the C-algebra with unit 1 with generators s i,...,s n 1 and relations s i s i+1 s i = s i+1 s i s i+1 (2.1) s i s j = s j s i if i j > 1 (2.2) s 2 i = 1 (2.3) Since every A-algebra can be seen as a module over itself, the investigation of an algebra A is equivalent to the study of its regular representation. We can generalize the notion of representations. 9

12 Definition 2.9. A representation of an associative algebra A over C with unit is a left A-module. The vector space is called the representation space and the bilinear map is called the representation map. Notice that the left and right module structures of the regular bimodule carry the same information. We can therefore limit our study to the left module structure to study representations in general. We will investigate how the left modules split up into smaller modules to find atomic modules i.e. modules that are undividable. We will make this formal through the following definitions. Definition Let A be an associative C algebra and let V be a left A-module. A vector space W V is called a submodule if a A, w W we have that a w W i.e. that the subvector space W is a left A-module itself for the bilinear map restricted to A W. Example For every module V the module itself and {0} are trivial submodules. Definition A module is called a simple module when its only submodules are itself and the zero module. Definition Let V be a left-a module with submodules W and U. If V is the direct sum of W, U as a vector space we say that V splits up, or decomposes, as a module as the direct sum of the modules W and U and write V = W U. Example For any module V we find the trivial decomposition V = V {0} = {0} V. Definition Let V be a module. We call V indecomposable, or irreducible, if for V = U W either V = {0} or W = {0} i.e. when the only decomposition of the module is the trivial decomposition. We have defined two types of smallest modules: those that don t split up into smaller modules and those that don t have true submodules. Algebras for which the two types of smallest representations coincide are called semisimple. Suppose that A reg decomposes into a direct sum of left A-modules M 1,..., M n A reg = n i=1 M i These subspaces M i A are called left ideals of A. 10

13 Definition A finite dimensional, associative, unital algebra A over C is called semisimple if A is the sum of its simple left ideals. Then the irreducible modules will coincide with the simples ones which enables us to study the regular representation of an algebra through its simple left modules. For a semisimple algebra the decomposition into left ideals is equivalent to a decomposition of the unit element. Suppose we have a decomposition A reg = n i=1 M i then we are able to find a decomposition of the unit element 1 = n e i where e i M i i=1 Definition A set {e i } A is called a complete set of mutually orthogonal idempotents of A when e i e j = δ ij e i and 1 = i e i. Example By considering the action of the element e i M i on the unit element of A e i = e i 1 = e i e j = e i e i j j we find that these e i in fact constitute a complete set of mutually orthogonal idempotents of A. Such a decomposition of the unit element is called a resolution of unity. Conversely, for v i M i we have v i = j e jv i such that e j v i = δ ij v i i.e. the idempotents e i act as projections onto the submodule M i = Ae i. So that through a resolution of unity we can find a corresponding decomposition of the module in submodules A reg = i Ae i Such a decomposition is called a left Peirce decomposition of the algebra A. Remark. A module M i is therefore indecomposable iff the corresponding idempotent cannot be resolved into a sum of nontrivial mutually orthogonal idempotents. In the case of indecomposability the idempotent is called a primitive idempotent. 11

14 Similarly we can define the right Peirce decomposition A = i e ia. Applying both at the same time gives the (two-sided) Peirce decomposition A = i,j e i Ae j In general we have the following situation for semisimple algebras. We have a complete set {e j i } of mutually orthogonal primitive components. Then A reg = i Where Ae j i = Ae j i as A-modules and Ae j i Ae j i as A-modules if i i. Also we find central elements j Ae j i z i = j e j i Z(A) called central idempotents, where 1 = i z i. We then define the so-called isotypical components through these central idempotents Az i = z i A = j e j i Example Let M n (C) be the matrix algebra of n n matrices over C. A linear basis in M n (C) is given by the set of n 2 matrix units: (e ij ) km := δ ik δ jm. The diagonal matrix units e ii, 1 i n are mutually orthogonal primitive idempotents as n I = e ii and e ii e jj = δ ij e ii i=1 The corresponding decomposition of the regular module is M n (C) = n i=1v i, where V i := M n (C)e ii. Notice that in this case the central idempotents coincide with the primitive idempotents. The following theorem by Wedderburn and Artin elucidates the profound connection between semisimple algebras and matrix algebras as it totally classifies the semisimple algebras as isomorphic to direct sums of matrix algebras. Theorem Wedderburn-Artin theorem Any semisimple C-algebra A which is finite dimensional as a vector space is isomorphic to an algebra of the type M n1 M n2... M ns for some integer s and an unordered set of integers {n 1,..., n s } called the numeric data of A. Under the isomorphism 12

15 the diagonal matrix units from the subalgebra M ni correspond to the primitive idempotents in A. Any A-module V is semisimple and can be uniquely presented as V = s i=1 k iv (i), where V (i) are pairwise non-isomorphic simple M ni -modules, these simple modules are called the blocks of the Algebra. In particular, simple A-modules correspond to V (i) i = 1,..., s and the regular module decomposes as A reg = s i=1 n iv (i) where n i = dim(v (i) ). Thus the numeric data can be identified with a set of dimensions of all the simple A-modules V (1), V (2),..., V (s) where n i = dim(v (i) ). Together with Maschke s theorem we obtain the well known fact of the reducibility of the group algebra of finite groups into simple left ideals. Theorem Maschke theorem. Let G be a finite group and K be a field. Then the group algebra K[G] is semisimple iff K is a field whose characteristic does not divide the order of G. In particular, the group algebra C[G] is semisimple. We shall conclude with a useful lemma taken from [3] to compute whether an algebra is semisimple. Definition The trace form of an algebra over a field of characteristic 0 is defined as the bilinear form (a, b) := T r(ρ a ρ b ), where ρ a : A A is the linear operator corresponding to an element a A defined by ρ a b = ab b A. When we write the trace form as a matrix D αβ in a C-basis, e α, of A we define the discriminant of A as the determinant of the matrix D αβ. Lemma A is semisimple iff its discriminant is non-zero. 2.3 Construction of the Hecke algebra For Coxeter groups W we can construct associative algebras over a commutative ring A (with 1) with a free A-basis parametrized by the elements of W. The multiplication law shall in a certain way reflect those of W. Furthermore the algebra depends on some parameters a s, b s A (s S), where a s = a t and b s = b t when t and s are conjugate in W. The starting point is a free A-module E on the set W, together with basis elements denoted T w (w W). The existence of an associative A-algebra structure on E is ensured by following theorem in Humphreys of which we shall omit the proof. 13

16 Theorem Given elements a s, b s as above, there exists a unique structure of associative A-algebra on the free A-algebra E, with T 1 acting as the identity, such that the following conditions hold s S, w W: T s T w = T sw if l(sw) > l(w), (2.4) T s T w = a s T w + b s T sw if l(sw) < l(w) (2.5) The algebra described in the theorem is denoted E A (a s, b s ) and is called a generic algebra. Note that for a s = 0, b s = 1 the algebra is the group algebra A[W ]. Remark. For a reduced expression w = s i1 s ir we find by induction that l(s ij s ir ) > l(s ij+1 s ir ) for i j i r s.t. by additional induction we find T w = T s1 T sr. Definition The Hecke algebra of a Coxeter group W is defined as the specialization of the generic algebra where A = C with a s = q 1 and b s = q for q C. The algebra is denoted H(q). The relations (2.4) and (2.5) then become T s T w = T sw if l(sw) > l(w), (2.6) T s T w = (q 1)T w + qt sw if l(sw) < l(w) (2.7) Remark. Note that T w H(q) is invertible, it suffices to show this for T s for s S, and (T s ) 1 = q 1 T s (1 q 1 )T Semisimplicity of the Hecke algebra for generic q In this section we shall state, without proof, when the Hecke algebra is semisimple. For a rigorous treatment the reader is advised to consult [3]. In the last section we defined the Hecke algebra as a sort of deformation of the group algebra C[W ]. Notice that for specialization q = 1 the Hecke algebra equals the group algebra: H(1) = C[W ]. However, more is true Theorem For q not equal to a root of unity the Hecke algebra, H(q) is semisimple. Furthermore the following isomorphism holds where W is the corresponding Coxeter group. H(q) = C[W ] (2.8) 14

17 The isomorphism is not canonical, but it thus turns out that the deformation for these generic q is not strong enough to break the semisimplicity of the group algebra. This is understood when we look at the group algebra as a matrix algebra; the deformation is rigid in the sense that the dimensions of the simple left ideals or matrix blocks don t change. Also when considering the discriminant of the Hecke algebra, we find that the discriminant, being a rational polynomial, cannot become 0 when q is not specialized to a root of unity. The Hecke algebra is therefore semisimple by lemma Representation theory of the Hecke algebra Representation theory preliminaries In this section we connect the theory of linear representations of finite groups, as discussed in [5], to the more general case of representations of associative algebras and modules. Definition Let V be a complex vector space. Then a linear representation of a finite group G with representation space V and representation map π is a group homomorphism π : G GL(V ) Example (i) For any finite group G, complex vector space V the representation map π : G GL(V ) g id V defines a representation. This is called the trivial representation. (ii) Let V be a complex vector space. We can extend the sign homomorphism defined in definition 2.3 to a representation. The representation map ε : S n GL(V ) σ ε(σ)id V defines a representation of the symmetric group. The representation is called the alternating representation. 15

18 Let π : G GL(V ) be a representation. A vector space W V which is G-invariant is called a subrepresentation. When a representation only has trivial subrepresentations, {0} and itself, the representation is called simple. If there are two subrepresentations U, W so that as vector spaces V = U W then we say V decomposes as a representation to the direct sum of U and W. If the only decomposition of the representation is the trivial one, V = {0} V = V {0}, we say the representation is irreducible, or indecomposable. The study of these irreducible representations is particularly important as is clear from the following theorem. Theorem Maschke theorem Every linear representation of a finite group is the direct sum of irreducible representations. Therefore the study of the representations of a group is reduced to the study of its irreducible representations. Definition Call two representations ρ : G GL(V ), ρ : G GL(V ) isomorphic if there exists a linear isomorphism τ : V V s.t. τ ρ = ρ τ. Write ρ = ρ. Definition The set of irreducible representations of G modulo isomorphism is denoted Ĝ. Theorem The number of irreducible representations of a finite group G equals the number of conjugacy classes of G. Thus #Ĝ = #Conjugacy classes of G It turns out that all the irreducible representations occur as subrepresentations of the so-called regular representation. Definition Define for a finite group G the regular representation by ρ(g)( g G ρ : G GL(C[G]) λ g e g ) g G λ g e gg Theorem The regular representation of a group G decomposes as the direct sum of multiples of the irreducible representations of that group, where the multiplicity of an irreducible representation in the decomposition equals 16

19 the dimension of the irreducible representation. Thus for irreducible representations π : G V π of dimension d π C[G] = π Ĝ V dπ π We are now able to connect these representations with the modules of the foregoing section. Remark. For G a finite group and V a complex vector space (i) If π : G GL(V ) is a representation, then V is a left C[G]-module with action C[G] V V, ( λ g e g, v) λ g π(g)v g G g G (ii) Conversely, a left C[G]-action (a, v) a v (a C[G]) on V gives rise to a representation π : G GL(V ) defined by π(g)(v) := e g v. The group algebra C[G] then becomes a left C[G]-module with action the multiplication in C[G] ( λ g e g, λ g e g ) = λ g ρ(g)( λ g e g ) = λ g λ g e gg g G g G g g g,g G Thus the module structure induced by the regular representation is exactly the regular representation C[G] reg from example 2.7 (ii) as it should be. The irreducible representations then correspond to simple modules. Now we have all the preliminaries to study the representation theory of the Hecke algebra of the symmetric group. Definition Define the Hecke algebra H n (q) as the Hecke algebra of the symmetric group S n, these are called Hecke algebras of type A n 1. By the theorem 2.26 H n (q) is semisimple for generic q and we have H n (q) = C[S n ]. The representation theory of the symmetric group is well understood and makes extensive use of partitions, Young diagrams and Young tableaux. The isomorphism 2.8 hints that the representation theory of the H n (q) can be done in an analogous manner. 17

20 2.4.2 Partitions, Young diagrams and tableaux Definition A partition λ = (λ 1, λ 2,...) of n, denoted λ n, is an ordered sequence of non-negative integers λ 1 λ 2..., such that i=1 λ i = n. The number of partitions of n is equal to the number of conjugacy classes in S n and by theorem 2.32 to the number of irreducible representations. The partitions are therefore a natural way to number the irreducible representations of S n. These partitions can be visualized as Young diagrams: diagrams composed of rows of boxes. For a partition λ the corresponding Young diagram, denoted [λ], has λ 1 boxes in the first row, λ 2 boxes in the second row, etc. Figure 2.1: The Young diagram corresponding to the partition λ = (5, 3, 1, 1) 10. Denote Λ n all Young diagrams with n boxes. We shall write µ < λ if µ can be obtained from λ by removing boxes. If µ < λ and µ has one respectively two boxes less than λ, µ will be denoted respectively λ (1), λ (2). Definition Let λ n define the conjugate partition λ by setting Lemma Let λ n (i) λ n. (ii) (λ ) = λ. λ i := #{j {1,..., n} λ j i} Proof. This is both immediate by realizing that λ λ reflecting the Young diagram in its main diagonal. For instance corresponds to =. By placing the numbers 1,2,...,n each number occurring once into the boxes of [λ] we obtain a Young tableau. 18

21 The box in which the number j is contained will be called the jth box. A Young tableau is called row-standard if the numbers in the rows are ascending left to right. Similarly a Young tableau is called columnstandard if the columns are ascending top to bottom. When the Young tableau is both column- and row-standard it s called a standard Young tableau Figure 2.2: An example of a standard Young tableau. These standard Young tableaux represent possible ways of building the corresponding Young diagram. We can inductively build every Young diagram with n boxes by adding a box to a Young diagram of n 1 boxes. The numbers in the tableau represent the step in which the box was added. In this way a standard Young tableaux corresponds to a path of Young diagrams. For instance for λ = (3, 2, 1) we have the Young diagram. A possible standard Young tableau is 6 which corresponds to the path P = {,,,,, } Note that the correspondence is a bijection as the rows and column of the tableau are ascending iff every diagram of the path is a Young diagram. Denote T λ all standard Young tableaux corresponding to λ n and T n all Young tableaux with n boxes. For λ Λ n and t T λ we define t (1) as the standard tableau t without the box containing n. Similarly t (2) is t without the boxes containing n and n 1. Definition (i) Define t λ the unique standard Young tableau where horizontally neighboring boxes contain succeeding numbers. (ii) Define t λ the unique standard Young tableau where vertically neighboring boxes contain succeeding numbers. The standard Young tableau in example is t λ for λ = (4, 3, 1) Young orthogonal representations of Hecke algebras of type A n Now we are able to construct irreducible representations for H n (q) for generic q, we shall thus assume from now on that q is never a root of unit, as 19

22 constructed by Hans Wenzl[4] as Hecke analogues of the Young orthogonal representations of the symmetric group. Specht Modules and Young orthogonal representations We will bring the representation theory of the symmetric group into remembrance as discussed in [6] and [7], for further reading see also [10]. We shall omit any proof but rather sketch how the Young s theory is developed and the irreducible representations, called Specht modules 1, are realized. Remember that we can write a permutation σ S n as a product of disjoint cycles. This product is unique up to order of cycles. After ordering the cycles in length in the product, we can associate a partition to the lengths of these cycles. For instance (123)(45), corresponds to λ = (3, 2). It turns out that this correspondence is unique up to conjugacy class. Therefore (123)(45) is not conjugate to (1234)(5) with cycle lengths {4, 1}. Of course for a permutation on n elements the total length of the disjoint cycles is n such that the conjugacy classes are numbered by the partitions of n. If we remember theorem 2.32 we thus find that the partitions of n are a natural way to parametrize the irreducible representations of S n. We can define an action of S n on T n by defining σt for T T n as the Young tableau with the numbers in the boxes of the tableau permutated by σ. Write Ω n = {1, 2,..., n}. For λ n every λ-tableau gives rise to a decomposition of Ω n into disjoint subsets Ω h i (T ) of cardinality λ i, where Ω h i (T ) is the collection of numbers in the ith row of T. Example The decomposition of Ω n for t λ, defined in definition 2.39 is given by {1,..., λ 1 } if i = 1 and λ 1 1, Ω h i (t λ ) = { i 1 k=1 λ k + 1,..., i k=1 λ k} if i > 1 and λ i 1, if λ i = 0 Definition Let λ n define S λ := {σ S n σ(ω h i (t λ )) = Ω h i (t λ ) 1}. i 1 After German mathematician Wilhelm Specht,

23 Now we are able to define the so-called Young projectors, for λ, µ n, H λ := σ S λ e σ C[S n ], (2.9) V µ := σ S µ ε(σ)e σ C[S n ] (2.10) where ε is the sign map defined in 2.3. Define w λ S n the unique element such that w λ t λ = t λ. Also define, letting λ n, the Young symmetrizer p λ := e (wλ ) 1V λ e w λ H λ C[S n ] Definition Define the Specht module M λ := C[S n ] p λ. Let λ n, define p λ := dim(m λ) n! p λ = dim(m λ) e (wλ ) n! 1V λ e w λ H λ C[S n ] (2.11) Theorem {p λ } λ Λn C[S n ] constitutes a complete set of mutually orthogonal primitive idempotents. Therefore we are ensured that we have found all irreducible representations and have a very concrete way of constructing these Specht modules. We conclude with the decomposition of the regular representation. Theorem The regular representation of the symmetric group S n decomposes as the direct sum of the irreducible Specht modules C[S n ] = λ n M dim(m λ) λ Wenzl s Hecke analogues For H n (q) the relations (2.6) and (2.7) for the generators T i (= T si ) become T i T i+1 T i = T i+1 T i T i+1 (2.12) T i T j = T j T i if i j > 1 (2.13) T 2 i = (q 1)T i + q (2.14) Note that when q = 1 we indeed get the same relations (2.1) - (2.3) as the relations for the generators of the group algebra C[S n ]. However it turns out 21

24 to be convenient to use projections g i (1 i n 1) to define the irreducible representations instead of T i. g i := q T i q + 1 H n(q) Then the relations (2.12) - (2.14) become g i g i+1 g i q (1 + q) g q 2 i = g i+1 g i g i+1 (1 + q) g 2 i+1 (2.15) g i g j = g j g i (2.16) g 2 i = g i (2.17) It can easily be shown that this is also a way of defining the Hecke algebra i.e. the Hecke algebra can be described as the associative algebra over C with unit generated by generators g 1,...,g n 1 with defining relations (2.15) - (2.17). For the definition of the representations we need rational functions a d for d Z \ {0} defined by a d (q) = 1 + q q d (1 + q)(1 + q q d 1 ) = 1 q d+1 (1 + q)(1 q d ) Lemma (i) a d + a d = 1 (ii) Let k, l, m N with k + m = l. Then and a d a d 1 = q (1 + q) 2 = a ka m + a l a m a k a l = a k a m + a l a k a m a l q (1+q) 2 Proof. The proofs of both are simple computations. We are now able to construct the representations. Remember S n acts on the Young tableaux by permuting the numbers in the boxes: for t T n the tableau s i (t) would be the tableau t where the numbers i and i + 1 interchanged. Definition Let V λ be the complex vector space with basis {v t, t T λ }. For simplicity of proofs we shall embed V λ into Ṽλ the complex vector space, whose basis is labeled by arbitrary Young tableaux of shape λ. The representations will be dependent on the relative position of successive numbers. Let t be a Young diagram, for a number i we write r i (t), c i (t) to denote 22

25 respectively the row, the column of t in which i is contained. We define a relative distance function for a Young tableau t. 2 d t,l,m := r l (t) r m (t) + c m (t) c l (t) By writing a t,l,m for a dt,l,m we define linear endomorphisms where π λ (g i ) : M (i) λ M (i) λ := Ṽλ t T λ :d t,i,i+1 0 and hence a t,i,i+1 (q) is well-defined. The endomorphism is defined by Cv t π λ (g i )v t = a t,i,i+1 (q)v t + (a t,i,i+1 (q)a t,i+1,i (q)) 1 2 vsi (t) (2.18) for a fixed suitable square root. Lemma For t a standard Young tableau d t,i,i+1 0 and hence V λ M (i) λ for all i. Proof. When d t,i,i+1 = 0 the box containing i + 1 must be on the diagonal below i. Thus if i is in the rth row, cth column for some r, c then i + 1 must be in the r + kth row, c + kth column for some k if d t,i,i+1 = 0. For instance, for k = 1 i j where j = i + 1, we have i < < i + 1 for both * s since t is standard, but this is of course impossible. For even larger separations (k 2) there are even more boxes in between. Remark. Notice that the lemma even holds in a wider sense and the same argumentation holds for i and i + 2, such that d t,i,i+2 0 for t a standard Young tableau. Proposition Let λ n. The linear endomorphisms π λ (e i ) induce a left H n (q)-module structure on V λ where the action by the generators g i on v t (t T λ ) is given by (g i, v t ) π λ (g i )v t V λ 2 Notice that we have a different notation from Wenzl. Where our d t,l,m is actually d t,l,m for Wenzl. This is because we follow Goodman s way to define representations for T L n (q). This amongst other things switches the Young diagram labels of the alternating and trivial representation. 23

26 Proof. By lemma 2.47 we know that the endomorphisms are well-defined, restricted from the M (i) λ to V λ. Step 1: π λ (g i )V λ V λ for all π λ (g i ). Consider when s i (t) is not a standard tableau. This only happens when i and i + 1 are in the same row or column since for other j we have j < i < i + 1 or i < i + 1 < j such that switching them preserves standardness. Since t is standard we then have d t,i,i+1 = 1 or d t,i+1,i = 1, such that a t,i,i+1 (q)a t,i+1,i (q) = a 1 a 1 = 1 0 = 0. Therefore π λ (g i )v t = a t,i,i+1 (q)v t V λ. Step 2: π λ respects the identities (2.15) - (2.17). Proving identity (2.17): g 2 i = g i Let t be any standard Young tableau and let V be the span of v t and v si (t). The action π λ (g i )v si (t) = a si (t),i,i+1(q)v si (t) + (a si (t),i,i+1(q)a si (t),i+1,i(q)) 1 2 vt is defined iff the action on v t is defined as a si (t),i,i+1(q) = a t,i+1,i (q) and a si (t),i+1,i = a t,i,i+1 (q). Hence V is invariant for the action of π λ (g i ). Let d = d t,i,i+1, then d si (t),i,i+1 = d t,i+1,i = d and π λ (g i ) V is given by the matrix ( a d (a d a d ) 1 2 (a d a d ) 1 2 a d ) (2.19) with respect to the basis {v t, v si (t)}, which is an idempotent by lemma 2.45 (i). Proving identity (2.16): g i g j = g j g i When i j > 1 we have s i s j (t) = s j s i (t) for any tableau with n boxes. Then it is easy to see with (2.18) that π λ (g i )π λ (g j )v t = π λ (g j )π λ (g i )v t. Proving identity (2.15): g i g i+1 g i Let t T n, i n 2 and also q (1+q) 2 g i = g i+1 g i g i+1 k = d t,i,i+1, l = d t,i,i+2 and m = d t,i+1,i+2 q (1+q) 2 g i+1 Where we know that k, l, m are non-zero by lemma 2.47 and remark such that a k, a l and a m are well-defined. Let p = s i1 s i2...s ij We have d si (t),r,s = d t,si (r),s i (t) for any t, r, s such that we have by induction d p(t),r,s = d t,p 1 r,p 1 s 24

27 We can use this to find, for instance, that d si s i+1 s i (t),i,i+1 = d t,i+2,i+1 = m and eventually obtain π λ (g i )π λ (g i+1 )π λ (g i )v t = (a 2 ka m + a k a k a l )v t + (a k a k ) 1 2 (ak a m + a k a l )v si (t) +a k (a m a m ) 1 2 al v si+1 (t) + a m (a k a k ) 1 2 (al a l ) 1 2 vsi+1 s i (t) +a k (a m a m ) 1 2 (al a l ) 1 2 vsi s i+1 (t) + (a k a k ) 1 2 (am a m ) 1 2 (al a l ) 1 2 vsi s i+1 s i (t) For π λ (g i+1 )π λ (g i )π λ (g i+1 )v t we obtain the same expression with i and i + 1 interchanged. It follows that On the other hand we have (π λ (g i )π λ (g i+1 )π λ (g i ) π λ (g i+1 )π λ (g i )π λ (g i+1 ))v t = (a 2 ka m + a k a k a l a 2 ma k a m a m a l )v t +(a k a k ) 1 2 (ak a m + a k a l a m a l )v si (t) +(a m a m ) 1 2 (ak a l a m a k a l a m )v si+1 (t) q (1 + q) (π λ(g 2 i ) π λ (g i+1 ))v t q ( ) = (a (1 + q) 2 k a m )v t + (a k a k ) 1 2 vsi (t) (a m a m ) 1 2 vsi+1 (t) It follows from lemma 2.45 (ii) that the expressions above are equal and hence the identity (2.15) holds. Example (i) Let λ = (1, 1, 1) 3. Thus λ =, there is only one standard Young tableau t = is one-dimensional. For therefore a t,2,1 = , such that V we find d t,1,2 = 1 such that a t,1,2 = a 1 = 0 and π (g 1 )v t = a t,1,2 v t + (a t,1,2 a t,2,1 ) 1 2 vsi (t) = a 1 v t + (a 1 a 1 ) 1 2 vsi (t) = 0v t + (1 0) 1 2 vsi (t) = 0 25

28 and hence π (T 1 )v t = π (q (q + 1)g 1 )v t = qv t (q + 1)π (g 1 )v t = qv t The computations for T 2 and T 3 are identical, therefore π (T i ) = q. We have found that π is isomorphic to the trivial representation. It is not hard to see that the computations are similar for λ = (1, 1,..., 1) n such that the trivial representation of H n (q) is isomorphic to the representation labeled by the diagrams with 1 column and n rows. (ii) Let λ = (3) 3. Thus λ =, there is only one standard Young tableau t = 1 2 3, such that V is one-dimensional. For we find d t,1,2 = 1 such that a t,1,2 = a 1 = 1 and therefore a t,2,1 = 0. π (g 1 )v t = a t,1,2 v t + (a t,1,2 a t,2,1 ) 1 2 vsi (t) = a 1 v t + (a 1 a 1 ) 1 2 vsi (t) = 1v t + (1 0) 1 2 vsi (t) = v t and hence π (T 1 )v t = π (q (q + 1)g 1 )v t = qv t (q + 1)π (g 1 )v t = v t The computations for T 2 and T 3 are identical, therefore π (T i ) = 1. We have found that π equals the alternating representation. It is not hard to see that the computations are similar for λ = (n) n such that the alternating representation of H n (q) is equal to the representation labeled by the diagrams with n columns and 1 row. Remark. We will make extensive use of the following notation λ (1) < λ. Let λ n. Then if we sum over λ (1) < λ we sum over the set {µ < λ µ n 1}. The elements of the set are therefore subdiagrams of λ with one box less. We want to define a bijection between T λ and λ (1) <λ T λ (1). For a standard Young Tableaux t T λ define the assignment t t (1), thus removing the nth box from t to find the diagram t (1) λ (1) <λ T λ (1). The inverse map is given 26

29 by adding a box to the underlying diagram of t (1) so that the underlying partition becomes λ. In the image of t (1) this new box of course is the box containing n. Example Let λ =. We find the following bijection We can extend this bijection to the vector spaces V λ, by defining the bijection on the basis by v t v t (1). We then find the isomorphism (as vector spaces) V λ = V λ (1) (2.20) λ (1) <λ By considering the algebra morphism defined by H n 1 (q) H n (q) g i g i we are able to realize H n 1 (q) as a subalgebra of H n (q) and it makes sense to restrict our H n (q) representations π λ to H n 1 (q) with representation space λ (1) <λ V λ (1). The isomorphism (2.20) commutes with the action of g 1,..., g n 2 and therefore with the action of elements of H n 1 (q) since the action only permutes the numbers of the first till n-1th box whereas the assignment t t (1) removes all nth boxes from the diagrams; the actions are disjoint. It does not matter whether you start with the action and end up with multiples of Young diagrams where the numbers 1,...,n-1 are permuted and then remove all nth boxes or if you first remove the nth box and then let the element act on your diagram to permute the numbers. We have found the following isomorphism π λ Hn 1 (q) = λ (1) <λ π λ (1) (2.21) Theorem Choose q generic. π λ as defined in the foregoing proposition is an irreducible representation of H n (q) on V λ for all λ Λ n. Furthermore 27

30 π λ = πµ iff λ = µ. The map π n : x H n (q) λ Λ n π λ (x) defines a faithful representation of H n (q). Proof. Step 1: irreducibility By theorem 2.26 we have that H n (q) is semisimple for all n N. We prove by induction. For n = 1 there is just one Young diagram and the Hecke algebra is just C. Remark. Note that in the following Young diagrams are labeled λ (1), λ (2), etc. This is only defined as an assignment for tableaux and is just a notational device in this context to keep track of how many boxes the diagram has in comparison with λ. Now suppose the proposition holds for H n 1 (q), remember that we can find central idempotents z λ (1) of the V λ (1). Again by using the identification T i T i for 1 i n 2 we find a way to H n 1 (q) H n (q). Using (2.20) and the definition of the central idempotents as projectors to the isotypical components we find V λ Hn 1 (q) = π λ (z λ (1))V λ λ (1) <λ and π λ (z λ (1))V λ = Vλ (1) Let 0 W V λ be a H n (q)-module, by the decomposition into irreducibles we must have for a λ (1) < λ V λ (1) W Hn 1 (q) If there is only one λ (1) < λ we are done as V λ = V λ (1) W. Otherwise we can choose a µ (1) another Young diagram with n 1 boxes such that µ (1) < λ. There exists exactly one λ (2) that is contained in both λ (1) and µ (1) : the one where both boxes are removed. Let t T λ be such that t (1) T λ (1) and t (2) T λ (2) and (s n 1 (t)) (1) T µ (1). With the isomorphism (2.20) and t (1) / T µ (1) we have v t (1) V λ (1) and µ (1) λ (1) and hence π λ (1)(z µ (1))v t (1) = 0. Therefore we find π λ (z µ (1))v t = 0 and putting this all together we get π λ (g n 1 )π λ (z µ (1))v t = π λ (z µ (1))π λ (e n 1 )v t = (a d a d ) 1 2 vsn 1 (t) V µ (1) where d = d t,n 1,n. We find that a d and a d 1 are well-defined and non-zero if we inspect how t is built. As we had two different subdiagrams it cannot 28

31 be the case that the nth box and the n-1th box touch in the tableau as you would then have an underlying Young diagram with a unique subdiagram: therefore d cannot equal 1 or -1. Furthermore as t is a standard Young tableau d cannot equal 0 either. Hence a d is well defined and not equal to 1,0 or -1 and we find that neither is a d using a d + a d = 1. Therefore V µ (1) W. As µ (1) was arbitrary V λ = λ (1) <λ V λ (1) W Hence V λ is irreducible. Step 2: π λ = πµ λ = µ For n = 1 there is just one Young diagram and for n = 2 we only have the trivial, and alternating representation which are of course nonisomorphic. Let n > 2 and λ, µ Λ n with µ λ. Then µ has a subdiagram µ (1) which is not a subdiagram of λ or vice versa. In the worst case scenario λ can be changed into µ by removing one box l of λ and adding a box m. Either µ or λ has more than one subdiagram (as unique subdiagrams only occur when the number of columns is a multiple of the number of rows or vice versa: has a unique subdiagram as well of course has multiple subdiagrams Suppose µ has two or more subdiagrams, than the subdiagram containing box m is the subdiagram of µ which isn t a subdiagram of λ. In better scenario s any subdiagram with one box less won t be a subdiagram of the other Young diagram. Therefore by isomorphism (2.20) V λ and V µ already differ as H n 1 (q)-modules and are therefore non-isomorphic as H n (q)-modules. Faithfulness Let ρ λ be the Young orthogonal representation of the symmetric group S n over C corresponding to λ. It is well known that ρ λ Sn 1 = λ <λ ρ λ thus by the decomposition (2.20) and induction we find that dim(ρ λ ) = dim(π λ ). 29

32 Therefore dim( λ Λ n π λ H n (q)) = dim(c[s n ]) = n! = dim(h n (q)) Which shows that π n is faithful. 30

33 Chapter 3 The Temperley-Lieb Algebra The Temperley-Lieb algebra is an algebra used throughout statistical physics, for instance as a connection between the symmetry classes of the ground states of the Hamiltonians of the XXZ spin chain and the combinatorics of alternating-sign matrices as discussed in [8]. The connection arises when the Temperley-Lieb algebra is studied through its action on matchings. The Temperley-Lieb algebra and matchings shall be discussed in the following section. The Temperley-Lieb algebra can be realized as a quotient of the Hecke algebra H n (q). Not only does this ensure semisimpleness for generic q, we are able to construct a complete set of irreducible representations for the Temperley-Lieb algebra by inquiring which Hecke representations descend to representations of the Temperley-Lieb algebra. It turns out that the irreducible representations labeled by a Young diagram with two rows or less constitute a complete set of irreducible representations of the Temperley-Lieb algebra. 3.1 The Temperley-Lieb algebra as an abstract algebra A p-matching of the vertex set [n] = {1,..., n} is an unordered collection of p disjoint pairs, or edges and n 2p single vertices. A matching is called crossing if it contains and edge {i, j} and a vertex k such that i < k < j or if it contains edges {i, j} and {k, l} such that i < k < j < l. Definition 3.1. A non-crossing perfect matching of [2n] is a noncrossing p-matching, where p = n. Denote F 2n the set of all non-crossing perfect matchings of [2n]. 31

34 Example 3.2. For n = 3 there are five non-crossing perfect matchings F 6 = {{1, 2}{3, 4}{5, 6}, {1, 4}{2, 3}{5, 6}, {1, 2}{3, 6}{4, 5}, {1, 6}{2, 3}{4, 5}, {1, 6}{2, 5}{3, 4}} An insightful way to visualize these matchings is by replacing the edges by loop segments connecting the vertices. Example 3.3. The matchings of F 6 are visualized as loops in the following way {1, 2}{3, 4}{5, 6} {1, 4}{2, 3}{5, 6} {1, 2}{3, 6}{4, 5} {1, 6}{2, 3}{4, 5} {1, 6}{2, 5}{3, 4} A third useful notation is a typographical notation using parentheses for paired vertices: F 6 = {()()(), (())(), ()(()), (()()), ((()))} We can build vector spaces over C, V F2, V F4,..., corresponding to the sets of perfect matchings F 2, F 4, etcetera. The basis vectors of the vector space are labeled by the perfect matchings. Example 3.4. (i) As there is only one perfect matching on [2]: V F2 := {λ () v () λ () C} 32

35 (ii) V F2n := { F F 2n λ F v F λ F C} We define matching generators or matchmakers e j j = 1,..., 2n 1 acting on elements F F 2n. A matchmaker e j only acts non-trivially on edges containing j or j + 1, two distinct situations arise: when there is an edge {j, j + 1} and otherwise. The action of e j is defined { {j, j + 1} {j, j + 1} e j : {i, j}{j + 1, k} {i, k}{j, j + 1} We can extend this action linearly to the vector spaces V F2n. Giving the vector spaces a left module structure for the associative algebra over C generated by the matchmakers. Proposition 3.5. The matchmakers e j 1 j 2n 1 satisfy the following relations: where q = e i 2π 3. e 2 j = (q + q 1 )e j, e j e j±1 e j = e j, e j e i = e i e j when i j > 1 Proof. The proof is immediate by realizing e i 2π 3 + e i 2π 3 = 1 and given the graphical representation below. The graphical representation of the e j is closely related to the graphical notation of the matchings of [2n], e j = 1 2 j-1 j j+1 j+2 2n-1 2n The action is given by placing the matchmaker beneath the matching. Example 3.6. The action of e 1 on ((())) is given by = 33