REPRESENTATIONS OF THE FRAMISATION OF THE TEMPERLEY–LIEB ALGEBRA

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OF THE TEMPERLEY–LIEB ALGEBRA

MARIA CHLOUVERAKI AND GUILLAUME POUCHIN

ABSTRACT. In this note, we describe the irreducible representations and give a dimension formula for the Framisation of the Temperley–Lieb algebra.

1. INTRODUCTION

The Temperley–Lieb algebra was introduced by Temperley and Lieb in [TeLi] for its applications in statistical mechanics. It was later shown by Jones [Jo1, Jo2] that it can be obtained as a quotient of the Iwahori–Hecke algebra of typeA. Both algebras depend on a parameterq. Jones showed that there exists a unique Markov trace, called the Ocneanu trace, on the Iwahori–Hecke algebra, which depends on a parameterz. For a specific value ofz, the Ocneanu trace passes to the Temperley–Lieb algebra. Jones used the Ocneanu trace on the Temperley–Lieb algebra to define a polynomial knot invariant, the famous Jones polynomial. Using the Ocneanu trace as defined originally on the Iwahori–Hecke algebra of typeAyields another famous polynomial invariant, the HOMFLYPT polynomial, which is also known as the2-variable Jones polynomial (the2variables beingqandz).

Yokonuma–Hecke algebras were introduced by Yokonuma as generalisations of Iwahori–

Hecke algebras in the context of finite Chevalley groups. The Yokonuma–Hecke algebra of typeAis the centraliser algebra associated to the permutation representation of the general linear group over a finite field with respect to a maximal unipotent subgroup. Juyumaya has given a generic presentation for this algebra, depending on a parameterq, and defined a Markov trace on it, the latter depending on several parameters [Ju1, JuKa, Ju2]. This trace was subsequently used by Juyumaya and Lambropoulou for the construction of invariants for framed knots and links [JuLa1, JuLa2]. They later showed that these invariants can be also used for classical and singular knots and links [JuLa3, JuLa4]. The next step was to construct an analogue of the Temperley–Lieb algebra in this case.

As it is explained in more detail in [JuLa5], where the technique of framisation is thor- oughly discussed, three possible candidates arose. The first candidate was the Yokonuma–

Temperley–Lieb algebra, which was defined in [GJKL1] as the quotient of the Yokonuma–

Hecke algebra by exactly the same ideal as the one used by Jones in the classical case. We studied the representation theory of this algebra and constructed a basis for it in [ChPo].

We are grateful to Dimoklis Goundaroulis, Jes´us Juyumaya, Aristides Kontogeorgis and Sofia Lambropoulou for introducing us to this whole range of problems, and for many fruitful conversations. We would also like to thank Tam´as Hausel for his interesting questions that led us to provide some extra results on the Yokonuma–Hecke algebra. The first author thanks the Centro di Ricerca Matematica Ennio De Giorgi in Pisa for its hospitality during the programme “Perspectives in Lie Theory” in January 2015. This research has been co-financed by the European Union (European Social Fund - ESF) and Greek national funds through the Operational Program

“Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: THALIS.

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The values of the parameters for which Juyumaya’s Markov trace passes to the Yokonuma–

Temperley–Lieb algebra are given in [GJKL1]. Unfortunately, for these values, the invariants for classical knots and links obtained from the Yokonuma–Temperley–Lieb algebra are equivalent to the Jones polynomial.

A second candidate, which is more interesting topologically, was suggested in [GJKL2].

This is the Framisation of the Temperley–Lieb algebra, whose representation theory we study in this paper. The Framisation of the Temperley–Lieb algebra is defined in a subtler way than the Yokonuma–Temperley–Lieb algebra, as the quotient of the Yokonuma–Hecke algebra by a more elaborate ideal, and it is larger than the Yokonuma–Temperley–Lieb algebra. The values of the parameters for which Juyumaya’s Markov trace passes to this quotient are given in [GJKL2]. It was recently shown that the Juyumaya–Lambropoulou invariants for classical links are stronger than the HOMFLYPT polynomial [CJKL]. It turns out that, in a similar way, the invariants for classical links obtained from the Framisation of the Temperley–Lieb algebra are stronger than the Jones polynomial.

The third candidate is the so-called Complex Temperley–Lieb algebra, which is larger than the Framisation of the Temperley–Lieb algebra, but provides the same topological information (see [JuLa5]).

In this note, we study the representation theory of the Framisation of the Temperley–

Lieb algebra. In Proposition 5 we give a complete description of its irreducible representations, by showing which irreducible representations of the Yokonuma–Hecke algebra pass to the quotient. The representations of the Yokonuma–Hecke algebra of typeAwere first studied by Thiem [Th1, Th2, Th3], but here we use their explicit description given later in [ChPA]. Our result generalises in a natural way the analogous result in the classical case.

We then use the dimensions of the irreducible representations of the Framisation of the Temperley–Lieb algebra in order to compute the dimension of the algebra. We deduce a combinatorial formula involving Catalan numbers, given in Proposition 6.

A reference to the results of this note is included in [GJKL2], so we decided to finally provide them in written form. We also take this opportunity to write down the relations between three types of generators used in bibliography so far (Remark 2), and show that the Yokonuma–Hecke algebra is split semisimple over a smaller field than the one considered in [ChPA] (Corollary 2).

2. THETEMPERLEY–LIEB ALGEBRA

In this section, we recall the definition of the Temperley–Lieb algebra as a quotient of the Iwahori–Hecke algebra of typeAgiven by Jones [Jo2], and some classical results on its representation theory.

2.1. The Iwahori–Hecke algebraHn(q). Letn∈Nand letqbe an indeterminate. The Iwahori–Hecke algebra of typeA, denoted byHn(q), is aC[q, q⁻¹]-associative algebra generated by the elements

G1, . . . , Gn−1

subject to the following relations:

(2.1) GiGj = GjGi for alli, j= 1, . . . , n−1with|i−j|>1, GiGi+1Gi = Gi+1GiGi+1 for alli= 1, . . . , n−2,

together with the quadratic relations:

(2.2) G²_i =q+ (q−1)Gi for alli= 1, . . . , n−1.

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Remark 1. If we specialiseqto1, the defining relations (2.1)–(2.2) become the defining relations for the symmetric groupSn. Thus the algebraHn(q)is a deformation of the group algebra overCofSn.

2.2. The Temperley–Lieb algebraTL_n(q). Leti= 1, . . . , n−1. We set G_i,i+1:= 1 +G_i+G_i+1+G_iG_i+1+G_i+1G_i+G_iG_i+1G_i.

We define theTemperley–Lieb algebraTL_n(q)to be the quotientH_n(q)/I, whereIis the ideal generated by the elementG1,2. We haveGi,i+1∈Ifor alli= 1, . . . , n−2, since

Gi,i+1= (G1G2· · ·G_n−1)ⁱ⁻¹G1,2(G1G2· · ·G_n−1)⁻⁽ⁱ⁻¹⁾.

2.3. Combinatorics of partitions. Letλ`nbe a partition ofn, that is,λ= (λ1, . . . , λk) is a string of positive integers such thatλ1≥λ2≥ · · · ≥λk ≥1and|λ|:=λ1+· · ·+λk= n. We shall also say thatλis a partitionof sizen.

We identify partitions with their Young diagrams: the Young diagram ofλis a left- justified array ofkrows such that thej-th row containsλjnodes for allj= 1, . . . , k. We writeθ= (x, y)for the node in rowxand columny.

For a nodeθlying in the linexand the columnyofλ(that is,θ = (x, y)), we define c(θ) :=q^y−x. The numberc(θ)is called the(quantum) contentofθ.

Now, atableau of shapeλis a bijection between the set{1, . . . , n}and the set of nodes inλ. In other words, a tableau of shapeλis obtained by placing the numbers1, . . . , nin the nodes ofλ. Thesizeof a tableau of shapeλisn, that is, the size ofλ. A tableau is standardif its entries increase along any row and down any column of the diagram ofλ.

For a tableauT, we denote byc(T |i)the quantum content of the node with the number iin it. For example, for the standard tableauT = 1 2 3 of size3, we have

c(T |1) = 1, c(T |2) =q and c(T |3) =q².

For any tableauT of sizenand any permutationσ∈Sn, we denote byT^σthe tableau obtained fromT by applying the permutationσon the numbers contained in the nodes of T. We have

c(T^σ|i) = c T |σ⁻¹(i)

for alli= 1, . . . , n.

Note that if the tableauT is standard, the tableauT^σis not necessarily standard.

2.4. Formulas for the irreducible representations ofC(q)Hn(q). We setC(q)Hn(q) :=

C(q)⊗_C[q,q⁻¹]Hn(q). LetP(n)be the set of all partitions ofn, and letλ∈ P(n). LetV_λ be aC(q)-vector space with a basis{v_T}indexed by the standard tableaux of shapeλ. We setv_T := 0for any non-standard tableauT of shapeλ. We have the following result on the representations ofC(q)Hn(q), established in [Ho]:

Proposition 1. Let T be a standard tableau of shape λ ∈ P(n). For brevity, we set c_i := c(T |i)fori = 1, . . . , n. The vector spaceV_λis an irreducible representation of C(q)Hn(q)with the action of the generators on the basis elementv_T defined as follows:

fori= 1, . . . , n−1,

(2.3) G_i(v_T) =qc_i+1−c_i+1 ci+1−ci

v_T +qc_i+1−c_i ci+1−ci

v_T_si ,

wheresiis the transposition(i, i+ 1). Further, the set{Vλ}_λ∈P(n)is a complete set of pairwise non-isomorphic irreducible representations ofC(q)Hn(q).

Corollary 1. The algebraC(q)Hn(q)is split semisimple.

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2.5. Irreducible representations ofC(q)TLn(q). Since the algebraC(q)Hn(q)is semisimple, the algebraC(q)TLn(q) := C(q)⊗_C_[q,q⁻¹_] TLn(q)is also semisimple. Moreover, the irreducible representations ofC(q)TLn(q)are precisely the irreducible representations of C(q)Hn(q) that pass to the quotient. That is, Vλ is an irreducible representation of C(q)TLn(q)if and only if G1,2(v_T) = 0for every standard tableauT of shapeλ. It is easy to check that the latter is equivalent to the trivial representation not being a direct summand of the restrictionRes^S_hsⁿ

1,s₂i(E^λ), whereE^λis the irreducible representation of the symmetric groupSn(equivalently, the algebraCHn(1)) labelled byλ. We thus obtain the following description of the irreducible representations ofC(q)TL_n(q):

Proposition 2. We have thatVλ is an irreducible representation of C(q)TLn(q)if and only if the Young diagram ofλhas at most two columns.

2.6. The dimension of C(q)TLn(q). Forn ∈ N, we denote by Cn the n-th Catalan number, that is, the number

Cn= 1 n+ 1

2n n

= 1

n+ 1

n

X

k=0

n k

² .

We have the following standard result on the dimension ofC(q)TLn(q)(cf. [Jo1, Jo2]):

Proposition 3. We have

dim_C(q)(C(q)TLn(q)) =Cn.

3. THE FRAMISATION OF THETEMPERLEY–LIEB ALGEBRA

In this section, we will look at a generalisation of the Temperley–Lieb algebra, which is obtained as a quotient of the Yokonuma–Hecke algebra of typeA. This algebra was introduced in [GJKL2], where some of its topological properties were studied. Here we will determine its irreducible representations and calculate its dimension.

3.1. The Yokonuma–Hecke algebraY_d,n(q). Letd, n∈N. Letqbe an indeterminate.

TheYokonuma–Hecke algebra of typeA, denoted byY_d,n(q), is a C[q, q⁻¹]-associative algebra generated by the elements

g1, . . . , g_n−1, t1, . . . , tn

subject to the following relations:

(3.1)

(b₁) gigj = gjgi for alli, j= 1, . . . , n−1with|i−j|>1, (b₂) gigi+1gi = gi+1gigi+1 for alli= 1, . . . , n−2,

(f₁) titj = tjti for alli, j= 1, . . . , n,

(f₂) tjgi = gits_i(j) for alli= 1, . . . , n−1andj= 1, . . . , n, (f₃) t^d_j = 1 for allj= 1, . . . , n,

wheresiis the transposition(i, i+ 1), together with the quadratic relations:

(3.2) g_i²=q+ (q−1)eigi for alli= 1, . . . , n−1, where

(3.3) ei:= 1

d

d−1

X

s=0

t^s_it^−s_i+1.

Note that we havee²_i =e_iande_ig_i=g_ie_ifor alli= 1, . . . , n−1. Moreover, we have (3.4) tiei=ti+1ei for alli= 1, . . . , n−1.

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Remark 2. In [ChPA], the first author and Poulain d’Andecy consider the braid generators

˜

gi:=q^−1/2giwhich satisfy the quadratic relation

(3.5) g˜²_i = 1 + (q^1/2−q^−1/2)ei˜gi.

On the other hand, in all the papers [Ju2, JuLa2, JuLa3, ChLa] prior to [ChPA], the authors consider the braid generators¯gi= ˜gi+(q^1/2−1)eig˜i(and thus,˜gi:= ¯gi+(q^−1/2−1)ei¯gi) which satisfy the quadratic relation

(3.6) ¯g²_i = 1 + (q−1)ei+ (q−1)eig¯i. Note that

(3.7) eigi=ei¯gi=q^1/2ei˜gi for alli= 1, . . . , n−1.

Remark 3. If we specialiseqto1, the defining relations (3.1)–(3.2) become the defining relations for the complex reflection groupG(d,1, n) ∼= (Z/dZ)oSn. Thus the algebra Yd,n(q) is a deformation of the group algebra overC of the complex reflection group G(d,1, n). Moreover, ford = 1, the Yokonuma–Hecke algebraY1,n(q)coincides with the Iwahori–Hecke algebraHn(q)of typeA.

Remark 4. The relations(b1),(b2),(f1)and(f2)are defining relations for the classical framed braid groupFn∼=ZoBn, whereBnis the classical braid group onnstrands, with thetj’s being interpreted as the “elementary framings” (framing 1 on thejth strand). The relationst^d_j = 1mean that the framing of each braid strand is regarded modulod. Thus, the algebraYd,n(q)arises naturally as a quotient of the framed braid group algebra over the modular relations(f₃)and the quadratic relations (3.2). Moreover, relations (3.1) are defining relations for the modular framed braid groupFd,n∼= (Z/dZ)oBn, so the algebra Yd,n(q)can be also seen as a quotient of the modular framed braid group algebra over the quadratic relations (3.2).

3.2. The Framisation of the Temperley–Lieb algebraFTLd,n(q). Leti= 1, . . . , n−1.

We set

g_i,i+1:= 1 +g_i+g_i+1+g_ig_i+1+g_i+1g_i+g_ig_i+1g_i.

We define theFramisation of the Temperley–Lieb algebrato be the quotientY_d,n(q)/I, whereIis the ideal generated by the element

e₁e₂g_1,2.

Note that, due to (3.4), e1e2 commutes withg1 and with g2, so it commutes withg1,2. Further, we havee_ie_i+1g_i,i+1∈Ifor alli= 1, . . . , n−2, since

e_ie_i+1g_i,i+1= (g₁g₂. . . g_n−1)ⁱ⁻¹e₁e₂g_1,2(g₁g₂. . . g_n−1)⁻⁽ⁱ⁻¹⁾.

Remark 5. In [GJKL2], the Framisation of the Temperley–Lieb algebra is defined to be the quotientY_d,n(q)/J, whereJis the ideal generated by the elemente₁e₂¯g_1,2, where

¯

g1,2= 1 + ¯g1+ ¯g2+ ¯g1¯g2+ ¯g2¯g1+ ¯g1¯g2g¯1.

Due to (3.7) and the fact that theei’s are idempotents, we havee1e2g¯1,2=e1e2g1,2, and soI=J.

Remark 6. Ford= 1, the Framisation of the Temperley–Lieb algebraFTL1,n(q)coincides with the classical Temperley–Lieb algebraTLn(q).

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3.3. Combinatorics ofd-partitions. Ad-partitionλ, or a Youngd-diagram, of sizenis a d-tuple of partitions such that the total number of nodes in the associated Young diagrams is equal ton. That is, we haveλ= (λ⁽¹⁾, . . . ,λ^(d))withλ⁽¹⁾, . . . ,λ^(d)usual partitions such that|λ⁽¹⁾|+· · ·+|λ^(d)|=n.

A pairθ = (θ, k)consisting of a nodeθand an integerk∈ {1, . . . , d}is called ad- node. The integerkis called thepositionofθ. Ad-partition is then a set ofd-nodes such that the subset consisting of thed-nodes having positionkforms a usual partition, for any k∈ {1, . . . , d}.

For ad-nodeθ lying in the linexand the columnyof thek-th diagram ofλ(that is, θ = (x, y, k)), we definep(θ) :=kandc(θ) :=q^y−x. The numberp(θ)is the position ofθand the numberc(θ)is called the(quantum) contentofθ.

Letλ = (λ⁽¹⁾, . . . ,λ^(d))be ad-partition ofn. Ad-tableau of shapeλis a bijection between the set{1, . . . , n}and the set ofd-nodes inλ. In other words, ad-tableau of shape λis obtained by placing the numbers1, . . . , nin thed-nodes ofλ. Thesizeof ad-tableau of shapeλisn, that is, the size ofλ. Ad-tableau isstandardif its entries increase along any row and down any column of every diagram inλ. Ford= 1, a standard1-tableau is a usual standard tableau.

For ad-tableauT, we denote respectively byp(T |i)andc(T |i)the position and the quantum content of thed-node with the number i in it. For example, for the standard 3-tableauT = 1 3 ,∅, 2

of size3, we have

p(T |1) = 1, p(T |2) = 3, p(T |3) = 1 and c(T |1) = 1, c(T |2) = 1, c(T |3) =q . For anyd-tableauT of sizenand any permutationσ ∈Sn, we denote byT^σthed- tableau obtained fromT by applying the permutationσon the numbers contained in the d-nodes ofT. We have

p(T^σ|i) = p T |σ⁻¹(i)

and c(T^σ|i) = c T |σ⁻¹(i)

for alli= 1, . . . , n.

Note that if thed-tableauT is standard, thed-tableauT^σis not necessarily standard.

3.4. Formulas for the irreducible representations ofC(q)Yd,n(q). The representation theory ofYd,n(q)has been first studied by Thiem in [Th1, Th2, Th3] and subsequently in [ChPA], where a description of its irreducible representations in terms ofd-partitions and d-tableaux is given.

LetP(d, n)be the set of alld-partitions ofn, and letλ∈ P(d, n). LetV˜λbe aC(q^1/2)- vector space with a basis{˜v_T}indexed by the standardd-tableaux of shapeλ. In [ChPA, Proposition 5], the authors describe actions of the generatorsg˜i, fori= 1, . . . , n−1, andtj, forj= 1, . . . , n, on{˜v_T}, which makeV˜λinto a representation ofYd,n(q)overC(q^1/2).

The matrices describing the action of the generatorst_j have complex coefficients, while the ones describing the action of the generators˜g_ihave coefficients inC(q^1/2). However, the change of basis

(3.8) v_T :=q^N^T^/2v˜_T,

whereNT := #{i∈ {1, . . . , n−1} |p(T |i)<p(T |i+ 1)}, and the change of generators

(3.9) gi=q^1/2g˜i

yield a description of the action ofYd,n(q)onV˜λwhich is realised overC(q)(see proposition below).

LetV_λbe aC(q)-vector space with a basis{v_T}indexed by the standardd-tableaux of shapeλ. We setv_T := 0for any non-standardd-tableauT of shapeλ. Let{ξ1, . . . , ξ_d}be

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the set of alld-th roots of unity (ordered arbitrarily). We setC(q)Yd,n(q) :=C(q)⊗_C_[q,q⁻¹_] Yd,n(q). The following result is [ChPA, Proposition 5] and [ChPA, Theorem 1], with the change of basis and generators described by (3.8) and (3.9).

Proposition 4. LetT be a standardd-tableau of shapeλ ∈ P(d, n). For brevity, we set pi := p(T |i)andci := c(T |i)fori = 1, . . . , n. The vector spaceVλis an irreducible representation ofC(q)Yd,n(q)with the action of the generators on the basis elementv_T defined as follows: forj= 1, . . . , n,

(3.10) tj(v_T) =ξp_jv_T ;

fori= 1, . . . , n−1, ifpi >pi+1then

(3.11) g_i(v_T) =v_T_si ,

ifp_i<p_i+1then

(3.12) gi(v_T) =qv_T_si ,

and ifpi= pi+1then

(3.13) gi(v_T) =qci+1−ci+1

ci+1−ci

v_T +qci+1−ci

ci+1−ci

v_T_si ,

wheresiis the transposition(i, i+ 1). Further, the set{Vλ}_λ∈P(d,n)is a complete set of pairwise non-isomorphic irreducible representations ofC(q)Yd,n(q).

Corollary 2. The algebraC(q)Y_d,n(q)is split semisimple.

Remark 7. Note that

(3.14) ei(v_T) =

0 , ifpi6= pi+1; v_T , ifpi= pi+1.

3.5. Irreducible representations ofC(q)FTL_d,n(q). Since the algebraC(q)Y_d,n(q)is semisimple, the algebraC(q)FTLd,n(q) :=C(q)⊗_C_[q,q−1]FTLd,n(q)is also semisimple.

Moreover, the irreducible representations ofC(q)FTL_d,n(q)are precisely the irreducible representations ofC(q)Y_d,n(q) that pass to the quotient. That is, V_λ is an irreducible representation ofC(q)FTLd,n(q)if and only ife1e2g1,2(v_T) = 0for every standardd- tableauT of shapeλ.

Proposition 5. We have thatV_λis an irreducible representation ofC(q)FTL_d,n(q)if and only if the Young diagram ofλ⁽ⁱ⁾has at most two columns for alli= 1, . . . , d.

Proof. Let us assume first thatVλis an irreducible representation ofC(q)FTLd,n(q)and leti ∈ {1, . . . , d}. Setn_i := |λ⁽ⁱ⁾|. Ifn_i 6 2, thenλ⁽ⁱ⁾ has at most two columns. If ni ≥3, let us consider all the standardd-tableauxT = (T⁽¹⁾, . . . ,T^(d))of shapeλsuch that

p₁= p₂= p₃=· · ·= p_n_i=i.

Then, using the notation of Proposition 1 for the Iwahori–Hecke algebraHn_i(q)and Equa- tion (3.14), we obtain

G1,2(v

T(i)) =g1,2(v_T) =g1,2e1e2(v_T) =e1e2g1,2(v_T) = 0

SinceT⁽ⁱ⁾runs over all the standard tableaux of shapeλ⁽ⁱ⁾, Proposition 2 yields thatλ⁽ⁱ⁾ has at most two columns.

Now assume that the Young diagram of λ⁽ⁱ⁾ has at most two columns for all i = 1, . . . , d. LetT = (T⁽¹⁾, . . . ,T^(d))be a standardd-tableau of shape λ. Ifp₁ = p₂ =

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p3=: p, then, by (3.14),e1e2g1,2(v_T) =g1,2e1e2(v_T) =g1,2(v_T). In this case,g1,2acts onv_T in the same way thatG1,2acts onv

T(p) (replacing the entries greater than3by entries in{4, . . . ,|λ^(p)|}). Following the result on the classical case, we haveg1,2(v_T) = 0.

Otherwise, again by (3.14), we havee₁e₂(v_T) = 0, ande₁e₂g_1,2(v_T) =g_1,2e₁e₂(v_T) =

0as desired.

3.6. The dimension ofC(q)FTL_d,n(q). We will now use the complete description of the irreducible representations ofC(q)FTL_d,n(q)by Proposition 5 to obtain a dimension formula forC(q)FTLd,n(q). Set

Kd,n:={(k1, k2, . . . , kd)∈N^d|k1+k2+· · ·+kd=n}.

Proposition 6. We have

dim_C_(q)(C(q)FTLd,n(q)) = X

(k1,k2,...,kd)∈Kd,n

n!

k₁!k₂!. . . k_d! ²

Ck₁Ck₂· · ·Ck_d.

Proof. Let us denote by P⁶²(d, n)the set of d-partitions λ of n such that the Young diagram ofλ⁽ⁱ⁾has at most two columns for alli= 1, . . . , d. By Proposition 5, and since the algebraC(q)FTLd,n(q)is semisimple, we have

λ∈P⁶²(d,n)

dim_C_(q)(Vλ)²,

wheredim_C_(q)(Vλ)is the number of standardd-tableaux of shapeλ.

Fix (k1, k2, . . . , kd) ∈ Kd,n. We denote by P⁶²(k1, k2, . . . , kd) the set of all d- partitionsλinP⁶²(d, n)such that|λ⁽ⁱ⁾|=k_ifor alli= 1, . . . , d. We have

(k1,k2,...,kd)∈Kd,n

X

λ∈P⁶²(k₁,k₂,...,k_d)

dim_C_(q)(Vλ)².

Letλ∈ P⁶²(k₁, k₂, . . . , k_d). We have n

k1

n−k1

k2

n−k1−k2

k3

· · ·

n−k1−k2− · · · −kd−1

kd

= n!

k1!k2!. . . kd! ways to choose the numbers in{1, . . . , n}that will be placed in the nodes of the Young diagram ofλ⁽ⁱ⁾for eachi= 1, . . . , d. We deduce that

dim_C_(q)(V_λ) = n!

k₁!k₂!. . . k_d!

d

Y

i=1

dim_C_(q)(V_λ(i)),

whereV_λ(i) is the irreducible representation ofC(q)TLk_i(q)labelled byλ⁽ⁱ⁾. We thus obtain thatdim_C_(q)(C(q)FTLd,n(q))is equal to

X

(k1,k2,...,kd)∈Kd,n

n!

k₁!k₂!. . . k_d! ²

X

λ∈P⁶²(k₁,k₂,...,k_d) d

Y

i=1

dim_C_(q)(V_λ(i))².

We now have that

X

λ∈P⁶²(k₁,k₂,...,k_d) d

Y

i=1

dim_C_(q)(V_λ(i))²

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is equal to

X

λ⁽¹⁾∈P⁶²(1,k1)

X

λ⁽²⁾∈P⁶²(1,k2)

. . . X

λ^(d)∈P⁶²(1,kd) d

Y

i=1

dim_C(q)(V_λ(i))²,

which in turn is equal to

d

Y

i=1





X

λ⁽ⁱ⁾∈P⁶²(1,k_i)

dim_C_(q)(V_λ(i))²



.

By Proposition 3, we have that X

λ⁽ⁱ⁾∈P⁶²(1,ki)

dim_C_(q)(V_λ(i))²= dim_C_(q)(C(q)TLk_i(q)) =Ck_i,

for alli= 1, . . . , d. We conclude that dim_C_(q)(C(q)FTL_d,n(q)) = X

(k1,k2,...,k_d)∈K_d,n

n!

k₁!k₂!. . . k_d! ²

C_k₁C_k₂· · ·C_k_d.

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