Matricial Rrepresentations of Certain Finitely Presented Groups Generated By Order-2 Generator and their Applications
aa r X i v : . [ m a t h . R T ] N ov MATRICIAL REPRESENTATIONS OF CERTAIN FINITELYPRESENTED GROUPS GENERATED BY ORDER-2GENERATORS AND THEIR APPLICATIONS
RYAN GOLDEN AND ILWOO CHO
Abstract.
In this paper, we study matricial representations of certain finitelypresented groups Γ N with N -generators of order-2. As an application, weconsider a group algebra A of Γ , under our representations. Specifically, wecharacterize the inverses g − of all group elements g in Γ , in terms of matricesin the group algebra A . From the study of this characterization, we realizethere are close relations between the trace of the radial operator of A , andthe Lucas numbers appearing in the Lucas triangle. Introduction
In this paper, we consider certain finite-dimensional representations ( C n , α n )of a finitely presented group Γ N , (1.0.1) Γ N def = (cid:10) { x , ..., x N } , { x = ... = x N } (cid:11) , where the generators x , ..., x N are noncommutative indeterminants . i.e., analgebraic structure Γ N is the group generated by N - generators x , ..., x N equippedwith its “noncommutative” binary operation ( · ), satisfying x = x = ... = x N = e N , where e N is the group-identity of Γ N such that: ge N = g = e N g, for all g ∈ Γ N , for all N ∈ N . So, one may understand a group Γ N as the quotient group ,(1.0.2) Γ N Group = F N / { g j = e : g j are generators of F N } , where F N is the noncommutative ( non-reduced ) free group (cid:10) { g j } Nj =1 (cid:11) with N -generators g , ..., g N , where e is the group-identity of the free group F N , and“ Group = ” means “being group-isomorphic” (e.g., [1] and [2]).By construction, all elements g of the group Γ N are of the form,(1.0.3) g = x k i x k i ... x k n i n , for some i , ..., i n ∈ { , ..., N } , and k , ..., k n ∈ Z , Mathematics Subject Classification.
Key words and phrases.
Finitely Presented Groups, Generators of Order 2, Representations,Matrix Groups, the Lucas Triangle, Lucas Numbers. as non-reduced words in { x , ..., x N } , where x − j means the group-inverse of x j , and hence, x − kj means (cid:0) x − j (cid:1) k , for all k ∈ N , for all j = 1 , ..., N. In this paper, for an arbitrarily fixed N ∈ N , we establish and study certain n -dimensional Hilbert-space representations ( C n , α n ) of the groups Γ N , for all n ∈ N \ { } . Under our representations, each element g of Γ N is understood as amatrix A g acting on C n , for all n ∈ N . Moreover, if g = x k i ...x k l i l as in (1.0.3)(as a “non-reduced” word in Γ N ), then there exists corresponding matrices A i , ..., A i l , such that A g = A k i ...A k l i l in M n ( C )(as a “reduced” word in M n ( C )), where M n ( C ) is the matricial algebra consistingof all ( n × n )-matrices, for all n ∈ N . Motivation.
The readers may skip reading this sub-section. Here, we simplywant to emphasize why we are interested in the groups Γ N , for N ∈ N . Free probability is a branch of operator algebra . By considering free-distributionaldata, one can establish operator-valued noncommutative probability theory on(topological, or pure-algebraic) algebras as in classical probability theory (e.g., [3],[4] and cited papers therein). Also, such free-probabilistic data let us have structuretheorems of given algebras under free product determined by given linear function-als . Here, the independence of classical probability is replaced by the so-called freeness . There are two approaches in free probability theory:
Voiculescu ’s originalanalytic approach (e.g., [4]), and
Speicher ’s combinatorial approach (e.g., [3]).The second-named author, Cho, has considered connections between operatoralgebra theory, and Hecke-algebraic number theory via free probability recently,to provide tools for studying number-theoretic results with operator-algebraic tech-niques, and vice versa, by establishing certain representational and operator-theoreticmodels from combinatorial free probability settings of [3].Especially, in the series of research papers [5], [2] and [1], Cho considered the free-probabilistic representations of the
Hecke algebras H ( G p ) generated by the generalized linear ((2 × matricial ) groups G p = GL ( Q p ) over the p - adic numberfields Q p , for primes p. In the frontier paper [5], Cho and Gillespie established free probability modelsof certain subalgebras H Y p of the Hecke algebras H ( G p ) by defining suitable linearfunctionals on H ( G p ) . In particular, they constructed free-probabilistic structurespreserving number-theoretic data from H Y p . In [2], Cho extended free-probabilistic models of H Y p in the sense of [6] to thoseof H ( G p ) fully, for all primes p. On such models, the C ∗ - algebras H ( G p ) were con-structed by realizing elements of H ( G p ) as operators under free-probabilistic repre-sentations of H ( G p ) . And the operator-theoretic properties of generating operatorsof H ( G p ) were studied there.In [1], by studying certain types of partial isometries of H ( G p ) , Cho obtained theembedded non-abelian multiplicative groups G N of H ( G p ) generated by N -manypartial isometries in the sense of [1]. In particular, it was shown there that:(1.1.1) G N Group = Γ N , for all N ∈ N , where Γ N is in the sense of (1.0.1), satisfying (1.0.2). ATRICIAL REPRESENTATIONS OF Γ N To study detailed algebraic properties of G N in H ( G p ) , and to investigate operator-algebraic properties of C ∗ ( G N ) in H ( G p ) , he used the isomorphic group Γ N of(1.0.1), and the corresponding group C ∗ -algebra C ∗ (cid:0) Γ N (cid:1) . Thus, for authors, it is natural to be interested in the groups Γ N . In this paper,we study the groups Γ N of (1.0.1) pure-algebraically (independent from those in[1], [2] and [5]).1.2. Overview.
In this paper, we concentrate on studying the groups Γ N of (1.0.1)independently. The main purpose is to establish suitable Hilbert-space representa-tions (other than those of [1]) of Γ N . Fundamentally, we construct “finite-dimensional”Hilbert-space representations ( C n , α n ) of Γ N , for “all” n ∈ N \ { } . Under our representation ( C n , α n ) for n ∈ N , each group element g ∈ Γ N isunderstood as a matrix A g in the matricial algebra M n ( C ) , consisting of all ( n × n )-matirices. To study an algebraic object g ∈ Γ N , we will investigate functionalproperties of the corresponding matrices A g ∈ M n ( C ) for g, for all n ∈ N (SeeSection 2).As application, we consider a group algebra A of Γ , under our representations.Specifically, we characterize the inverses g − of all group elements g in Γ , in termsof matrices in the group algebra A generated by Γ (See Section 3).From the study of this characterization of Section 3, we show that there are closerelations between the trace of the powers T n of the radial operator T of A , andthe Lucas numbers in the
Lucas triangle (See Section 4).2.
Finite-Dimensional Representations of Γ N In this section, we establish finite-dimensional Hilbert-space representations ofthe finitely presented groups Γ N with N -generators of order-2 in the sense of (1.0.1).i.e., Γ N = (cid:10) { x j } Nj =1 , { x j = e N } Nj =1 (cid:11) , where x , ..., x N are noncommutative indeterminants (as generators), and e N are the group-identities of Γ N , for all N ∈ N . Fix N ∈ N throughout this section and the corresponding group Γ N . A Matrix Group M ( N ) . Let M ( C ) be the (2 × C × = C \ { } in C . Suppose C × C × is the Cartesian product of C and C × . Define now a function A : C × C × → M ( C )by(2.1.1) A ( a, b ) = (cid:18) a b − a b − a (cid:19) , for all ( a, b ) ∈ C × C × . It is not difficult to check that:
Lemma 2.1.
Let A be the map in the sense of (2.1.1). Then(2.1.2) ( A ( a, b )) = I , the identity matrix of M ( C ) , for all ( a, b ) ∈ C × C × . RYAN GOLDEN AND ILWOO CHO
Proof.
The proof of (2.1.2) is from the straightforward computation.It is also easy to verify that: A ( a , b ) A ( a , b ) = a a b + b − b a b a b − b a a b − a b a − a b + a b a b b b − b a + a a b b ! , and A ( a , b ) A ( a , b ) = b − b a + a a b b b a − a b − a b + a b a + a b − a b a b b a a b + b − b a b ! , and hence,(2.1.3) A ( a , b ) A ( a , b ) = A ( a , b ) A ( a , b ) , in general, for ( a k , b k ) ∈ C × C × , for k = 1 , . In particular, whenever( a k , b k ) ∈ C × × C × in C × C × , for k = 1 , , and the pairs ( a , b ) and ( a , b ) are “distinct” in C × × C × , theabove noncommutativity (2.1.3) always holds.Therefore, the images { A ( a, b ) : ( a, b ) ∈ C × × C × } forms the noncommutative family in M ( C ) , by (2.1.3).Now, take distinct pairs,(2.1.3) ′ ( a k , b k ) ∈ C × × C × , for k = 1 , ..., N. For the above chosen N -pairs { ( a k , b k ) } Nk =1 of (2.1.3) ′ , one can construct thecorresponding matrices,(2.1.4) A ( a k , b k ) in M ( C ) , for k = 1 , ..., N, by (2.1.1). Under the matrix multiplication on M ( C ) , let’s construct the mul-tiplicative subgroup M ( N ) of M ( C ) as(2.1.5) M ( N ) = (cid:10) { A ( a k , b k ) } Nk =1 (cid:11) generated by A ( a k , b k ) , for all k = 1 , ..., N. i.e., all elements of M ( N ) are the(2 × M ( C ) , satisfying(2.1.6) ( A ( a k , b k )) = I in M ( N ) , for all k = 1 , ..., N, be (2.1.2). Definition 2.1.
Let M ( N ) be the multiplicative subgroup (2.1.5) of M ( C ) . Wecall M ( N ) , the -dimensional, order-2, N -generator (sub-)group (of M ( C ) ). One obtains the following algebraic characterization.
Theorem 2.2.
Let M ( N ) be a 2-dimensional, order-2, N -generator group in M ( C ) . Then the groups M ( N ) and the group Γ N of (1.0.1) are isomorphic. i.e.,(2.1.7) M ( N ) Group = Γ N . ATRICIAL REPRESENTATIONS OF Γ N Proof.
Let M ( N ) be a 2-dimensional, order-2, N -generator group in M ( C ) . Then,by (2.1.6), each generator A ( a k , b k ) satisfies( A ( a k , b k )) = I , for all k = 1 , ..., N. i.e., each generator A ( a k , b k ) has order-2 indeed in M ( N ) . Since ( a k , b k ) are taken from C × × C × , the generators { A ( a k , b k ) } Nk =1 forms anoncommutative family in M ( C ) , by (2.1.3).Observe now that there does not exist n -tuple ( j , ..., j n ) of “distinct” elements j , ..., j n in { , ..., N } , for all n ∈ N , such that n Π l =1 A (cid:16) a k jl , b k jl (cid:17) = A (cid:0) a k j , b k j (cid:1) , or I , for some j ∈ { , ..., N } , by the very definition (2.1.1) and (2.1.3).These observations show that M ( N ) Group = F N / R N , where F N is the noncommutative free group (cid:10) { g j } Nj =1 (cid:11) with N -generators g , ..., g N , and R N means the relator set, R N = { g j = e } Nj =1 , where e is the group-identity of F N . Recall that the group Γ N of (1.0.1) is group-isomorphic to the quotient group F N / R N , by (1.0.2). Therefore, one has that M ( N ) Group = F N / R N Group = Γ N . Therefore, the 2-dimensional, order-2, N -generator group M ( N ) is group-isomorphicto the group Γ N of (1.0.1).By (2.1.7), two groups M ( N ) and Γ N are isomorphic from each other, equiva-lently, there does exist a group-isomorphism, α : Γ N → M ( N ) , satisfying(2.1.8) α ( x j ) = A ( a j , b j ) , for all j = 1 , ..., N. The above structure theorem (2.1.7), equivalently, the existence of the group-isomorphism α of (2.1.8) provides a 2-dimensional Hilbert-space representation (cid:0) C , α (cid:1) of Γ N . Theorem 2.3.
There exists the -dimensional Hilbert-space representation (cid:0) C , α (cid:1) of Γ N . Especially, α ( g ) acting on C are in the sense of (2.1.8), for all g ∈ Γ N .i.e.,(2.1.9) (cid:0) C , α (cid:1) forms a 2-dimensional representation of Γ N , where α is in the sense of (2.1.8).Proof. Since α of (2.1.8) is a generator-preserving group-isomorphism from Γ N onto M ( N ) , it satisfies α ( g g ) = α ( g ) α ( g ) in M ( N ) , for all g , g ∈ Γ N , and RYAN GOLDEN AND ILWOO CHO α (cid:0) g − (cid:1) = (cid:0) α ( g ) (cid:1) − in M ( N ) , for all g ∈ Γ N . Since M ( N ) ⊆ M ( C ) , the matrices α ( g ) are acting on C , for all g ∈ Γ N . Therefore, the pair (cid:0) C , α (cid:1) forms a Hilbert-space representation of Γ N . Certain Order-2 Matrices in M n ( C ) . Now, let n > N . And considercertain types of matrices in M n ( C ) . In Section 2.1, we showed that our group Γ N of (1.0.1) is group-isomorphic to the multiplicative subgroup M ( N ) of (2.1.5) in M ( C ) , by (2.1.7), and hence, we obtain a natural 2-dimensional representation (cid:0) C , α (cid:1) of Γ N in (2.1.9). In this section, we construct a base-stone to extendthe representation (cid:0) C , α (cid:1) to arbitrary n -dimensional representations ( C n , α n )of Γ N , for all n ∈ N \ { } . To do that, we fix the matrices formed by A ( a, b ) = (cid:18) a b − a b − a (cid:19) , as blocks of certain matrices in M n ( C ) . Assumption
For our main purpose, we always assume that:( a, b ) ∈ C × ˜1 × C × , whenever we have matrices A ( a, b ) in the following text, as in (2.1.3) ′ , where C × ˜1 def = C \ { , } . (cid:3) Assume first that n = 3 , and consider a matrix A formed by(2.2.1) A = c − bc − a a b − a b − a ∈ M ( C ) , for some a, b, c ∈ C × . As we have seen in (2.2.1), this matrix A is regarded asthe following block matrix,(2.2.1) ′ A = (1) (cid:0) c − bc − a (cid:1)(cid:18) (cid:19) A ( a, b ) in M ( C ) , where A ( a, b ) is in the sense of (2.1.1).If we have (3 × A in the sense of (2.2.1) understood as (2.2.1) ′ , then,from straightforward computation, one can obtain that(2.2.2) A = I = , the identity matrix of M ( C ) . Let’s define now a morphism A : C × ˜1 × C × × C × → M ( C ) by ATRICIAL REPRESENTATIONS OF Γ N (2.2.3) A ( a, b, c ) = (1) (cid:0) c − bc − a (cid:1)(cid:18) (cid:19) A ( a, b ) , identified with the matrix (2.2.1) in M ( C ) , by (2.2.1) ′ . By (2.2.2), for any ( a, b, c ) ∈ C × ˜1 × C × × C × , we have( A ( a, b, c )) = I in M ( C ) . Now, let n = 4 in N , and consider a matrix A formed by(2.2.3) A = d − bd − a c − bc − a a b − a b − a ∈ M ( C ) , for c, d ∈ C × . Then, similarly, this matrix A of (2.2.3) is regarded as the blockmatrix,(2.2.4) A = (1) (cid:0) d − bd − a (cid:1) A ( a, b, c ) , or A = I d − bd − a c − bc − a O , A ( a, b ) . in M ( C ) , where O , is the (2 × zero matrix whose all entries are zeroes.Then, by the direct computation, one obtains that(2.2.5) A = I , the identity matrix of M ( C ) . So, similar to (2.2.3), we define a morphism A : C × ˜1 × ( C × ) → M ( C ) by(2.2.6) A ( a, b, c, d ) = I Q a,b ( c, d ) O , A ( a, b ) , where I is the (2 × O , is the (2 × Q a,b ( c, d ) = d − bd − a c − bc − a .The image A ( a, b, c, d ) of (2.2.6) becomes a well-defined matrix in M ( C ) , by(2.2.3) and (2.2.4). Moreover, by (2.2.5), we have(2.2.7) RYAN GOLDEN AND ILWOO CHO ( A ( a, b, c, d )) = I in M ( C ) . Let’s consider one more step: let n = 5 . Similarly, we define the following map, A : C × ˜1 × ( C × ) → M ( C ) by(2.2.8) A ( a , ..., a ) = I Q a ,a ( a , a , a ) O , A ( a , a ) , in M ( C ) , for all ( a , ..., a ) ∈ ( C × ) , where A ( a , a ) is in the sense of (2.1.1), O , is the (2 × I is the (3 × Q a ,a ( a , a , a ) = a − a a − a a − a a − a a − a a − a . i.e., A ( a , ..., a ) = a − a a − a a − a a − a a − a a − a a a − a a − a ∈ M ( C ) . From direct computation, one again obtain that(2.2.9) ( A ( a , ..., a )) = I , the identity matrix of M ( C ) . Inductively, for n ≥ , we define the following map A n : C × ˜1 × ( C × ) n − → M n ( C ) , by(2.2.10) A n ( a , ..., a n ) = I n − Q a ,a ( a , ..., a n ) O ,n − A ( a , a ) , for all ( a , ..., a n ) ∈ ( C × ) n , where A ( a , a ) is in the sense of (2.1.1), O ,n − is the (2 × ( n − I n − is the ( n − × ( n −
2) identity matrix,and Q a ,a ( a , ..., a n ) = a n − a a n − a ... ... a − a a − a a − a a − a . Then we obtain the following computation.
ATRICIAL REPRESENTATIONS OF Γ N Theorem 2.4.
Let A n be the morphism (2.2.10), and let A n ( a , ..., a n ) be theimage of A n realized in the matricial algebra M n ( C ) , for arbitrarily fixed n ≥ in N . Then(2.2.11) ( A n ( a , ..., a n )) = I n , the identity matrix of M n ( C ) . Proof.
Let n ≥ N . Then, by (2.2.2), (2.2.7) and (2.2.9), we have( A k ( a , ..., a k )) = I k in M k ( C ) , for k = 3 , , . Now, without loss of generality take n ≥ N , generally. Then, by (2.2.10), A n ( a , ..., a n ) = I n − Q a ,a ( a , ..., a n ) O ,n − A ( a , a ) . For convenience, we let I denote = I n − , Q denote = Q a ,a ( a , ..., a n ) , and O denote = O ,n − , A denote = A ( a , a ) . i.e., A n ( a , ..., a n ) denote = (cid:18) I QO A (cid:19) , as a block matrix in M n ( C ) . Then( A n ( a , ..., a n )) = (cid:18) I QO A (cid:19) (cid:18)
I QO A (cid:19) = I + QO IQ + QAOI + AO OQ + A (2.2.12) = I IQ + QAO A = I Q + QAO I , by (2.1.2).So, to show (2.2.11), it is sufficient to show that Q + QA = O n − , , by (2.2.12).Notice that QA = a n − a a n − a ... ... a − a a − a a − a a − a a a − a a − a ! = a a n + − a a n (1 − a ) a (1 − a ) a a n + a a a n − a ... ... a a + − a a (1 − a ) a (1 − a ) a a + a a a − a = − a n a a n (cid:16) a − a (cid:17) ... − a a a (cid:16) a − a (cid:17) . Thus, Q + QA = a n − a a n − a ... ... a − a a − a + − a n a a n (cid:16) a − a (cid:17) ... − a a a (cid:16) a − a (cid:17) = a a n (cid:16) − − a + 1 + a − a (cid:17) ... ...0 a a (cid:16) − − a + 1 + a − a (cid:17) = = O n − , , i.e., Q + QA = O n − , . So, for any n ≥ , (2.2.13)( A n ( a , ..., a n )) = I Q + QAO I = (cid:18) I O n − , O I (cid:19) = I n , the identity matrix of M n ( C ) . Therefore, for any n ≥ N , ( A n ( a , ..., a n )) = I n in M n ( C ) . ATRICIAL REPRESENTATIONS OF Γ N n -Dimensional Hilbert-Space Representations of Γ N . Let N ∈ N be thefixed quantity, and fix n ∈ N \ { } . Also, let Γ N be our finitely presented group(1.0.1). For the fixed n , we take the n -tuples(2.3.1) W k = ( a k, , ..., a k,n ) ∈ C × ˜1 × ( C × ) n − , for k = 1 , ..., N, and assume that W , ..., W N are “mutually distinct.” Observation and Notation
As we assumed above, let W , ..., W N be mutuallydistinct in C × ˜1 × ( C × ) n − . For our purposes, one may further assume that thesemutually distinct n -tuples satisfy W i ≇ W j in the sense that: a i,l = a j,l in C × , for all l = 1 , ..., N. If the n -tuples W , ..., W N satisfy the above stronger condition than the mutually-distinctness, we say they are strongly mutually distinct . (cid:3) Define the corresponding matrices,(2.3.2) X k denote = A n ( W k ) = I n − Q a k, ,a k, ( a k, , ..., a k,n ) O , n − A ( a k, , a k, ) , as in (2.2.10), where W k are “strongly mutually distinct” n -tuples of (2.3.1), forall k = 1 , ..., N. Then, as in Section 2.1, one can get the multiplicative subgroup M n ( N ) of M n ( C )by the (reduced) free group generated by { X , ..., X N } . i.e.,(2.3.3) M n ( N ) = (cid:10) { X j } Nj =1 (cid:11) in M n ( C ) , where X j ’s are in the sense of (2.3.2), for all j = 1 , ..., N. Definition 2.2.
We call the multiplicative subgroup M n ( N ) of M n ( C ) , an n -dimensional, order-2, N -generator (sub-)group (of M n ( C ) ). Then, similar to the proof of the structure theorem (2.1.7), one can obtain thefollowing generalized result.
Theorem 2.5.
Let M n ( N ) be an n -dimensional, order-2, N -generator group in M n ( C ) . Then it is group-isomorphic to the group Γ N of (1.0.1). i.e.,(2.3.4) M n ( N ) Group = Γ N , for all n ∈ N \ { } . In particular, there exist generator-preserving group-isomorphisms α n : Γ N → M n ( N ) , such that(2.3.5) α n ( x j ) = X j , for all j = 1 , ..., N, where x j ’s are the generators of Γ N , and X j ’s are in the sense of (2.3.2). Proof.
Let n = 2 . Then, by (2.1.7), the isomorphic relation (2.3.4) holds. Assumenow that n ≥ . Then the proof of (2.3.4) is similar to that of (2.1.7). Indeed, onecan show that M n ( N ) Group = F N / R N Group = Γ N , for all n ∈ N . Therefore, there exists a natural generator-preserving group-isomorphism α n from Γ N onto M n ( N ) as in (2.3.5).The structure theorem (2.3.4) is the generalized result of (2.1.7). And the group-isomorphism α n of (2.3.5) generalizes α of (2.1.8). Therefore, we obtain the fol-lowing theorem, generalizing (2.1.9). Theorem 2.6.
Let Γ N be the group in the sense of (1.0.1), for some N ∈ N . Thenthere exist n -dimensional Hilbert-space representation ( C n , α n ) of Γ N , where α n are in the sense of (2.3.5), for all n ∈ N \ { } . i.e.,(2.3.6) ( C n , α n ) form n -dimensional representations of Γ N , for all n ∈ N . Proof. If n = 2 , as we have seen in (2.1.9), there exists a 2-dimensional Hilbert-space representation (cid:0) C , α (cid:1) , where α is the group-isomorphism in the sense of(2.1.8).Suppose n ≥ N . For such n, two groups Γ N and M n ( N ) are isomorphicwith a group-isomorphism α n : Γ N → M n ( N ) of (2.3.5), by (2.3.4). It shows thatthe images α n ( g ) are ( n × n )-matrices acting on the n -dimensional Hilbert space C n . So, the pair ( C n , α n ) forms an n -dimensional representation of Γ N , for all n ∈ N . The above two theorems show that, for a fixed group Γ N of (1.0.1), one has asystem { ( C n , α n ) } ∞ n =2 of Hilbert-space representations, and the corresponding isomorphic groups (cid:8) M n ( N ) (cid:9) ∞ n =2 , acting on C n (or, acting in M n ( C )). Example 2.1.
Let Γ be the finitely presented group, Γ = (cid:10) { x , x } , { x = x } (cid:11) . Fix n = 3 . Now, take the following strongly distinct triples, W = ( t , t , t ) , W = ( s , s , s ) , in C × ˜1 × ( C × ) , and construct two matrices, A ( W ) = t − t t − t t t − t t − t , A ( W ) = s − s s − s s s − s s − s , in M ( C ) . The group M (2) is established as the reduced free group h{ A ( W ) , A ( W ) }i generated by A ( W ) and A ( W ) in M ( C ) . Then M (2) Group = Γ . ATRICIAL REPRESENTATIONS OF Γ N So, one has a natural -dimensional representation (cid:0) C , α (cid:1) , where α is thegroup-isomorphism satisfying α ( x l ) = A ( W l ) , for all l = 1 , . Application: A Group Algebra A Induced by Γ In Section 2, we showed that the finitely presented group Γ N of (1.0.1) can haveits family of finite-dimensional Hilbert-space representations { ( C n , α n ) } ∞ n =2 , since it is group-isomorphic to n -dimensional, order-2, N -generator groups M n ( N )in M n ( C ) , for all n ∈ N \ { } , where M n ( N ) = (cid:10) { A n ( W k ) } Nk =1 (cid:11) in M n ( C ) , where W , ..., W N are strongly mutually distinct n -tuples of C × ˜1 × ( C × ) n − , forall n ∈ N \ { } . Algebraic Observation for Group Elements of Γ . In this section, weconcentrate on the case where N = 2 , i.e., we study the group elements of Γ indetail. As a special case of (2.3.4),(3.1.1) (cid:10) { x , x } , { x = x = e } (cid:11) = Γ
22 Group = M (2) . Consider Γ pure-algebraically. Each element g of Γ has its expression,(3.1.2) g = x k i x k i · · · x k in i n , for some n ∈ N , as in (1.0.3), for some ( i , ..., i n ) ∈ { , } n , and ( k , ..., k n ) ∈ Z n . However, by the relation on Γ ,x = e denote = e = x in Γ , one has that(3.1.3) x − l = x l , for l = 1 , , and x k +1 l = x l , for all k ∈ N . In other words, the generators x and x are “self-invertible,” and “indeed oforder-2,” in Γ . The above two conditions in (3.1.3) can be summarized by(3.1.3) ′ x n +1 l = x l , for l = 1 , , for all n ∈ Z . So, the general expression (3.1.2) of g is in fact(3.1.4) g = x j x j ...x j n in Γ , for some ( j , ..., j n ) ∈ { , } n , for some n ∈ N , by (3.1.3), or by (3.1.3) ′ . By the characterization, Γ
22 Group = F / R , and by the definition of the noncommutative (non-reduced) free group F , theexpression (3.1.4) of g goes to(3.1.5) g = e = e or x or x or( x x ) n x or( x x ) n or( x x ) n x or( x x ) n , in Γ , for all n ∈ N . Proposition 3.1.
Let g ∈ Γ . Then g is only one of the forms in (3.1.5). (cid:3) Let g ∈ Γ \ { e, x , x } . Say, g = x x x x ...x x x . Then it is self-invertible. Indeed, g = ( x x ...x x x ) ( x x ...x x x )= x x ...x x (cid:0) x (cid:1) x ...x x x = x x ...x x ex ...x x x = x x ...x (cid:0) x (cid:1) x ...x x x = ... = x = e, i.e.,(3.1.6) (( x x ) n x ) = e, for all n ∈ N . Similarly, one obtains that(3.1.7) (( x x ) n x ) = e in Γ , for all n ∈ N . Now, let g = x x x x ...x x in Γ . Then g − = x x x x ...x x , with | g | = (cid:12)(cid:12) g − (cid:12)(cid:12) in N , where | w | means the length of w ∈ Γ . (For example, if g = x x x x x ∈ Γ , then | g | = 5 . )Indeed, gg − = ( x x ...x x ) ( x x ...x x ) = e. So, one has that(3.1.8) (( x x ) n ) − = ( x x ) n , and equivalently,(3.1.9) (( x x ) n ) − = ( x x ) n , ATRICIAL REPRESENTATIONS OF Γ N for all n ∈ N . Proposition 3.2.
Let g ∈ Γ \ { e } , and let | g | means the length of g in { x , x } in Γ . (3.1.10) If | g | is odd in N , then g − = g in Γ . (3.1.11) If | g | is even in N , then g = ( x x ) k , or ( x x ) k , and g − = ( x x ) k , respectively, ( x x ) k , for some k ∈ N . Proof.
Suppose | g | is odd in N . Then, by (3.1.5), the group element g is one of x , or x , or ( x x ) n x , or ( x x ) n x , for n ∈ N . By (3.1.6) and (3.1.7), in such cases, g = e in Γ . So, the statement (3.1.10) holds.Now, assume that | g | is even in N . Then, by (3.1.5), this element g is either( x x ) k , or ( x x ) k , for k ∈ N . By (3.1.8) and (3.1.9), one has that (cid:16) ( x x ) k (cid:17) − = ( x x ) k , respectively, (cid:0) ( x x ) k (cid:1) − = ( x x ) k , for all k ∈ N . Therefore, the statement (3.1.11) holds, too.By the self-invertibility of the group-identity e, the generators x , x , and thegroup elements g with odd length | g | , we are interested in the cases where | g | is even.Equivalently, we are interested in the cases where g = ( x x ) k , or g = ( x x ) k in Γ . Analytic-and-Combinatorial Observation for Elements of Γ in M ( C ) . As we have seen, the group Γ is group-isomorphic to the 2-dimensional, order-2,2-generator subgroup M (2) of M ( C ) under our representation (cid:0) C , α (cid:1) . In par-ticular, M (2) is generated by two matrices A ( t , s ) , and A ( t , s )for the strongly mutually distinct pairs ( t , s ) and ( t , s ) in C × × C × . Moreprecisely, A ( t , s ) = t s − t s − t ! , and A ( t , s ) = t s − t s − t ! in M ( C ) . Notation
In the rest of this paper, we denote A ( t , s ) and A ( t , s ) , by X and X , respectively, for convenience. (cid:3) As we mentioned at the end of Section 3.1, we are interested in group-elements g formed by(3.2.1) g = ( x x ) k ∈ Γ , for k ∈ N . i.e., under α , if g is not of the form of (3.2.1), we obtain a self-invertible matrix α ( g ) , such that (cid:0) α ( g ) (cid:1) = I in M (2) ⊂ M ( C ) . However, if g is of (3.2.1) in Γ , then(3.2.2) α ( g ) = α (cid:0) ( x x ) k (cid:1) = ( X X ) k , and hence, (cid:0) α ( g ) (cid:1) − = α (cid:0) g − (cid:1) = ( X X ) k , in M (2) . Observe that(3.2.3) ( X X ) + ( X X ) = (cid:16) t t + s (1 − t ) s + s (1 − t ) s (cid:17) I from the straightforward computation in M ( C ) . Consider now that det ( X ) = − X ) , and det ( X + X ) = det t + t s + s − t s + − t s − t − t ! = − (cid:0) t + 2 t t + t (cid:1) − (cid:18) − t + s ( − t ) s + s ( − t ) s + 1 − t (cid:19) = − t − t t − t − t − s ( − t ) s − s ( − t ) s − t = − t t − s ( − t ) s − s ( − t ) s − , i.e. det ( X + X ) = − − (cid:18) t t + s ( − t ) s + s ( − t ) s (cid:19) , where det( · ) means the determinant on M ( C )So, one can get that(3.2.4)det ( X ) + det ( X ) − det ( X + X ) = 2 t t + s ( − t ) s + s ( − t ) s . ATRICIAL REPRESENTATIONS OF Γ N Lemma 3.3.
Let X and X be the generating matrices of M (2) as above. Thenthere exists ε ∈ C , such that(3.2.5) ( X X ) + ( X X ) = ε I in M ( C ) , and ε = 2 t t + s ( − t ) s + s ( − t ) s = det ( X ) + det ( X ) − det ( X + X ) . Proof.
The proof of (3.2.5) is done by (3.2.3) and (3.2.4).Therefore, by (3.2.5), one can observe that( X X + X X ) = ( X X ) + X X X X + X X X X + ( X X ) = ( X X ) + ( X X ) + 2 I , and hence, ( X X ) + ( X X ) = ( X X + X X ) − I = ( ε I ) − I = (cid:0) ε − (cid:1) I , where ε is in the sense of (3.2.5). i.e., we obtain that(3.2.6) ( X X ) + ( X X ) = (cid:0) ε − (cid:1) I . By (3.2.5) and (3.2.6), we have that( X X ) + ( X X ) = ε I , and ( X X ) + ( X X ) = (cid:0) ε − (cid:1) I , where ε = det( X ) + det( X ) − det ( X + X ) , as in (3.2.5).More generally, we obtain the following recurrence relation. Theorem 3.4.
Let X and X be the generating matrices of M (2) in M ( C ) . Ifwe denote X k denote = ( X X ) k + ( X X ) k , for all k ∈ N , then the following recurrence relation is obtained:(3.2.7) X = ε I , and X = (cid:0) ε − (cid:1) I , and X n = ε X n − − X n − , for all n ≥ in N , in M ( C ) , where ε = det ( X ) + det( X ) − det( X + X ) . Proof.
By (3.2.5), indeed, one has X = ε I , and by (3.2.6), X = (cid:0) ε − (cid:1) I . Suppose n = 3 in N . Then X = ( X X ) + ( X X ) = ( X X + X X ) − (3 X X + 3 X X )by (3.1.5) = ( X ) − X = ( ε I ) − ε I )= (cid:0) ε − ε (cid:1) I in M ( C ) . Observe now that ε X − X = ε (cid:0) ε − (cid:1) I − ε I = (cid:0) ε − ε (cid:1) I − ε I = (cid:0) ε − ε (cid:1) I , in M ( C ) . Thus, one obtains that(3.2.8) X = (cid:0) ε − ε (cid:1) I = ε X − X . So, if n = 3 , then the relation (3.2.7) holds true.Assume now that the statement(3.2.9) X n = ε X n − − X n − hold for a fixed n ≥ N . Then X n +1 = (( X X ) n + ( X X ) n ) ( X X + X X ) − ( X X ) n ( X X ) − ( X X ) n ( X X )= (( X X ) n + ( X X ) n ) ( ε I ) − (cid:0) ( X X ) n − + ( X X ) n − (cid:1) by the assumption (3.2.9)= ε X n − X n − , i.e., for the fixed n ≥ , (3.2.10) X n +1 = ε X n − X n − . Since n is arbitrary in N \ { , } , we can conclude that X n +1 = ε X n − X n − ,for all n ∈ N \ { } , by the induction, because of (3.2.8), (3.2.9) and (3.2.10).Therefore, we obtain the recurrence relation: X = ε I , X = (cid:0) ε − (cid:1) I , and X n = ε X n − − X n − , for all n ≥ N . Equivalently, the relation (3.2.7) holds.
ATRICIAL REPRESENTATIONS OF Γ N By the recurrence relation (3.2.7), we obtain that: X X + X X = ε I , ( X X ) + ( X X ) = (cid:0) ε − (cid:1) I , ( X X ) + ( X X ) = (cid:0) ε − ε (cid:1) I . and ( X X ) + ( X X ) = ε (cid:0) ε − ε (cid:1) I − (cid:0) ε − (cid:1) I = (cid:0) ε − ε + 2 (cid:1) I , in M ( C ) . Inductively, one can verify that:
Corollary 3.5.
There exists a functional sequence ( f n ) ∞ n =1 , such that(3.2.11) ( X X ) n + ( X X ) n = ( f n ( ε )) I , in M ( C ) . (cid:3) For instance, if ( f n ) ∞ n =1 is in the sense of (3.2.11), then f ( z ) = z, f ( z ) = z − , f ( z ) = z − z, and f ( z ) = z − z + 2 , etc.The following theorem provides the refined result of (3.2.11). Theorem 3.6.
Let X and X be the generating matrices of a -dimensional,order-2, 2-generator subgroup M (2) of M ( C ) in the sense of (3.2.1), and let X l = ( X X ) l + ( X X ) l ∈ M ( C ) , for all l ∈ N . Then there exists a functional sequence ( f n ) ∞ n =1 such that X n = f n ( ε ) I in M ( C ) , where ε = det( X ) + det( X ) − det( X + X ) . Moreover,(3.2.12) f n ( z ) = P [ n ] k =0 ( − k nk (cid:18) n − k − k − (cid:19) z n − k = P [ n ] k =0 ( − k nn − k (cid:18) n − kk (cid:19) z n − k , where [ n ] = the maximal integer ≤ n , and (cid:18) ml (cid:19) = m ! l !( m − l )! , for all m, l ∈ N , for all n ∈ N . Proof.
By (3.2.7) and (3.2.11), there exists a functional sequence ( f n ) ∞ n =1 such that X n = f n ( ε ) I in M ( C ) , for all n ∈ N , where ε = det( X ) + det( X ) − det( X + X ) . For instance,(3.2.13) f ( z ) = z, f ( z ) = z − , and f ( z ) = z − z, etc. So, it suffices to show that each n -th entry of the sequence f n ( z ) satisfies(3.2.12), for all n ∈ N . Say n = 1 . Then the function f ( z ) of (3.2.12) satisfies f ( z ) = P [ ] k =0 ( − k nn − k (cid:18) n − k − k − (cid:19) z n − k = P k =0 ( − k nn − k (cid:18) n − kk (cid:19) z n − k = ( − − (cid:18) − (cid:19) z − = z, satisfying(3.2.14) f ( z ) = z. If n = 2 , then the function f ( z ) of (3.2.12) goes to f ( z ) = P k =0 ( − k nn − k (cid:18) n − kk (cid:19) z n − k = ( − − (cid:18) − (cid:19) z − + ( − − (cid:18) − (cid:19) z − = z − , satisfying(3.2.15) f ( z ) = z − . Therefore, the formula (3.2.12) holds true where n = 1 , , by (3.2.13).Now, we consider the functions f n ( z ) of (3.2.12) satisfy the recurrence relation;(3.2.16) f ( z ) = z, f ( z ) = z − , and f n +1 ( z ) = zf n ( z ) − f n − ( z ) , for all n ≥ N . First of all, the functions f and f satisfy the initial condition of the relation(3.2.16), by (3.2.14) and (3.2.15). So, concentrate on the cases where n ≥ N . Observe that zf n ( z ) − f n − ( z ) ATRICIAL REPRESENTATIONS OF Γ N = z (cid:18)P [ n ] k =0 ( − k nk (cid:18) n − k − k − (cid:19) z n − k (cid:19) − P [ n − ] k =0 ( − k n − k (cid:18) n − k − k − (cid:19) z n − − k = P [ n ] k =0 ( − k nk (cid:18) n − k − k − (cid:19) z n − k +1 − P [ n +12 ] k =1 ( − k − n − k − (cid:18) n − k − k − (cid:19) z n − − k +2 = P [ n ] k =0 ( − k nk (cid:18) n − k − k − (cid:19) z n +1 − k + P [ n +12 ] k =1 ( − k n − k − (cid:18) n − k − k − (cid:19) z n +1 − k = z n +1 + P [ n ] k =1 ( − k nk (cid:18) n − k − k − (cid:19) z n +1 − k + P [ n +12 ] k =1 ( − k n − k − (cid:18) n − k − k − (cid:19) z n +1 − k = z n +1 + P [ n ] k =1 ( − k (cid:18) nk (cid:18) n − k − k − (cid:19) + n − k − (cid:18) n − k − k − (cid:19)(cid:19) z n +1 − k + ( − [ n +12 ] n − n +12 ] − (cid:18) n − [ n +12 ] − n +12 ] − (cid:19) z n +1 − n +12 ] = z n +1 + P [ n ] k =1 ( − k (cid:16) n ( n − k − k ( n − k − − k +1)!( k − + ( n − n − k − k − n − k + k +2)!( k − (cid:17) z n +1 − k + Y where(3.2.17) Y = ( − [ n +12 ] n − n +12 ] − (cid:18) n − [ n +12 ] − n +12 ] − (cid:19) z n +1 − n +12 ] , and then= z n +1 + P [ n ] k =1 ( − k (cid:16) n ( n − k − n − k )! k ! + n ( n − k − − ( n − k − n − k +1)!( k − (cid:17) z n +1 − k + Y = z n +1 + P [ n ] k =1 ( − k (cid:16) ( n − k +1) n ( n − k − kn ( n − k − − k ( n − k − k ( n − k +1)!( k − (cid:17) z n +1 − k + Y = z n +1 + P [ n ] k =1 ( − k (cid:16) ( n − kn + n + nk − k )( n − k − k ( n − k +1)!( k − (cid:17) z n +1 − k + Y = z n +1 + P [ n ] k =1 ( − k (cid:16) ( n − nk + n − k )( n − k − k ( n − k +1)!( k − (cid:17) z n +1 − k + Y = z n +1 + P [ n ] k =1 ( − k (cid:16) ( n +1)( n − k )( n − k − k ( n − k +1)!( k − (cid:17) z n +1 − k + Y = z n +1 + P [ n ] k =1 ( − k n +1 k ( n − k )!( n − k − ( k − k − z n +1 − k + Y = z n +1 + P [ n ] k =1 ( − k n +1 k (cid:18) n − kk − (cid:19) z n +1 − k + Y = P [ n ] k =0 ( − k n +1 k (cid:18) n − kk − (cid:19) z n +1 − k + Y , i.e., for any n ≥ , we obtain(3.2.18) zf n ( z ) − f n − ( z ) = P [ n ] k =0 ( − k n +1 k (cid:18) n − kk − (cid:19) z n +1 − k + Y , where Y is in the sense of (3.2.17).Notice that if n ≥ N , then[ n ] = [ n +12 ] in N , and hence, the first term of (3.2.18) contains the second one. Therefore, one has(3.2.19) n ∈ N ⇒ zf n ( z ) − f n − ( z ) = P [ n +12 ] k =0 ( − k n +1 k (cid:18) n − kk − (cid:19) z n +1 − k , if and only if n ∈ N = ⇒ zf n ( z ) − f n − ( z ) = f n +1 ( z ) . Now, assume n ≥ N . Then Y = ( − [ n +12 ] (cid:16) n − n +1 − (cid:17) (cid:18) ( n − [ n +12 ] − ) ! ( n − [ n +12 ] − − [ n +12 ]+2 ) ! ( [ n +12 ] − ) ! (cid:19) z n +1 − n +12 ] = ( − [ n +12 ] (cid:16) n − n − (cid:17) (cid:18) ( n − [ n +12 ] − ) ! ( n − n +12 ]+1 ) ! ( [ n +12 ] − ) ! (cid:19) z . = ( − [ n +12 ] (2) (cid:18) ( n − [ n +12 ] ) ! ( n − n +12 ]+1 ) ! ( n − [ n +12 ] )( [ n +12 ] − ) ! (cid:19) z . = ( − [ n +12 ] (cid:16) n +1[ n +12 ] (cid:17) (cid:16) [ n +12 ] − n − [ n +12 ] (cid:17) n − [ n +12 ][ n +12 ] − z . = ( − [ n +12 ] (cid:16) n +1[ n +12 ] (cid:17) (cid:16) n +1 − n − n − (cid:17) n − [ n +12 ][ n +12 ] − z . = ( − [ n +12 ] (cid:16) n +1[ n +12 ] (cid:17) n − [ n +12 ][ n +12 ] − z . i.e.,(3.2.20) ATRICIAL REPRESENTATIONS OF Γ N n ∈ N + 1 = ⇒ Y = ( − [ n +12 ] (cid:16) n +1[ n +12 ] (cid:17) n − [ n +12 ][ n +12 ] − z . So, by (3.2.20), if n ≥ N , then(3.2.21) zf n ( z ) − f n − ( z ) = P [ n +12 ] k =0 ( − k n +1 k (cid:18) n − kk − (cid:19) z n +1 − k , if and only if zf n ( z ) − f n − ( z ) = f n +1 ( z ) . Therefore, by (3.2.19) and (3.2.21), one can conclude that(3.2.22) n ≥ N = ⇒ zf n ( z ) − f n − ( z ) = f n +1 ( z ) . The above result (3.2.22) shows that f n +1 ( ε ) = ε f n ( ε ) − f n − ( ε ) , for n ≥ N , with f ( ε ) = ε , and f ( ε ) = ε − . Recall that, by (3.2.7), X = ε I , and X = (cid:0) ε − (cid:1) I , and X n +1 = ε X n − X n − = f n ( ε ) I . Therefore, one can conclude that if f n ( z ) = P [ n ] k =0 ( − k (cid:0) nk (cid:1) (cid:18) n − k − k − (cid:19) z n − k , then X n = f n ( ε ) I , for all n ∈ N . By the recurrence relation (3.2.7), we obtain the above main theorem of Section3.
Summary of Section 3.2 If X and X are the generating matrices of the2-dimensional, order-2, 2-generator subgroup M (2) of M ( C ) , then(3.2.23) ( X X ) n + ( X X ) n = f n ( ε ) I , in M ( C ) , where f n ( z ) = P [ n ] k =0 ( − k (cid:0) nk (cid:1) (cid:18) n − k − k − (cid:19) z n − k , and ε = det( X ) + det( X ) − det( X + X ) , for all n ∈ N . (cid:4) Let x and x be the generators of the group Γ of (1.0.1), and let α ( x ) = X and α ( x ) = X be the corresponding generating matrices of the isomorphic group M (2) of Γ in M ( C ) . The main result (3.2.23) shows that the generators x and x of Γ inducesthe analytic data in M ( C ) depending on the elements x x and its inverses x x up to (3.2.23).3.3. Group Algebras A α n , Γ N . Let Γ be an arbitrary discrete group, and let (
H,α ) be a
Hilbert-space representation of
Γ consisting of a Hilbert space H and thegroup-action α : Γ → L ( H ) , making α ( g ) : H → H, for all g ∈ Γbe linear operators (or linear transformations) on H, satisfying α ( g g ) = α ( g ) α ( g ) , for all g , g ∈ Γ , and α ( g − ) = ( α ( g )) − , for all g ∈ Γ , where L ( H ) means the operator algebra consisting of all linear operators on H. Remark that, if we are working with topologies , then one may replace L ( H )to B ( H ) , the operator algebra consisting of all bounded (or continuous ) linearoperators on H. However, we are not considering topologies throughout this paper,so we simply set the representational models for L ( H ) here.For example, our group Γ N of (1.0.1) has its Hilbert-space representations ( C n , α n ) , for all n ∈ N \ { } , and each element g of Γ N is realized as α n ( g ) ∈ M n ( N ) in(3.3.1) M n ( C ) = L ( C n ) = B ( C n ) . The first equivalence relation (=) of (3.3.1) holds because, under finite-dimensionality,all linear operators on C n are matrices acting on C n , and vice versa. The secondequivalence relation (=) of (3.3.1) holds, because, under finite-dimensionality, allpure-algebraic operators on C n are automatically bounded (and hence, continuous).For an arbitrary group Γ , realized by ( H, α ) , one obtains the isomorphic group α (Γ) in L ( H ) , i.e., Γ Group = α (Γ) in L ( H ) . So, one can construct a subalgebra,(3.3.2) A α, Γ = C [ α (Γ)] of L ( H ) , where C [ X ] mean the polynomial ring in sets X. Such rings C [ X ] form algebrasunder polynomial addition and polynomial multiplication over C , and hence, un-der the inherited operator addition and operator multiplication, C [ α (Γ)] forms analgebra in L ( H ) in (3.3.2). ATRICIAL REPRESENTATIONS OF Γ N Definition 3.1.
Let Γ be a group and let ( H, α ) be a Hilbert-space representationof Γ . The subalgebra A α, Γ of L ( H ) in the sense of (3.3.2) is called the group algebraof Γ induced by ( H, α ) . It is not difficult to show that if(3.3.3) A o Γ = C [Γ] , the pure-algebraic algebra generated by Γ , as a polynomial ring in Γ , then(3.3.4) A o Γ Alg = A α, Γ ,for all Hilbert-space representations ( H, α ) of Γ algebraically, where “
Alg = ” means“being algebra-isomorphic.”
Proposition 3.7.
Let Γ N be the group of (1.0.1), and let ( C n , α n ) be our n -dimensional representations of Γ N , for all n ∈ N \ { } . Let M n ( N ) be the n -dimensional, order-2, N -generator subgroups of M n ( C ) , for all n ∈ N \ { } . If A α n , Γ N are in the sense of (3.3.2), and A Γ N is the group algebra in the sense of(3.3.3), then(3.3.5) A Γ N Alg = A α n , Γ N = C (cid:2) M n ( N ) (cid:3) , for all n ∈ N . Proof.
The proof of (3.3.5) is from the general relation (3.3.4). Recall that, by thegroup-isomorphism α n ,α n (cid:0) Γ (cid:1) = M n ( N ) , for all n ∈ N . A Group Algebra A α , Γ . In this section, we take the group algebra in thesense of (3.3.2),(3.4.1) A α , Γ denote = A ,as a subalgebra of M ( C ) = L ( C ), under our representation (cid:0) C , α (cid:1) . By(3.3.5), pure-algebraically, the algebra A of (3.4.1) is algebra-isomorphic to thegroup algebra A Γ = C [Γ ] . By (3.1.5), all elements of A are the linear combinations of the matrices formedby (3.4.2) I , X , X , ( X X ) n , ( X X ) n , or ( X X ) n X , ( X X ) n X , for all n ∈ N , where X j = α ( x j ), for j = 1 , , since A = C (cid:2) M (2) (cid:3) in M ( C ) . As we have seen in Section 3.1, the building blocks of A in the sense of (3.4.2)satisfy that: (i) I , X , X , ( X X ) n X , and ( X X ) n X are self-invertible in A ⊂ M ( C ) , and (ii) (( X X ) n ) − = ( X X ) n , for all n ∈ N . Corollary 3.8.
Let X and X be the generating matrices of M (2) , and hence,those of A . Then(3.4.3) (( X X ) n ) − = f n ( ε ) I − ( X X ) n , and (( X X )) − = f n ( ε ) I − ( X X ) n for all n ∈ N , where ( f n ) ∞ n =2 is the functional sequence in the sense of (3.2.12).Proof. By (3.2.23), we have( X X ) n + ( X X ) n = f n ( ε ) I in A , where f n are in the sense of (3.2.12), and ε is in the sense of (3.2.5), for all n ∈ N . So, ( X X ) n = f n ( ε ) I − ( X X ) n in A , for all n ∈ N . Thus,(( X X ) n ) − = f n ( ε ) I − ( X X ) n , for all n ∈ N . Similarly, we obtain(( X X ) n ) − = f n ( ε ) I − ( X X ) n , for all n ∈ N . It means that if T is a matrix of A in the form of either ( X X ) n or ( X X ) n , for any n ∈ N , then T − = f n ( ε ) I − T. Proposition 3.9.
Let A ∈ M (2) in A . Then(3.4.4) A − = (cid:26) A or r A I − A, for some r A ∈ C . Proof. If A ∈ M (2) in A , then T is one of the forms of (3.4.2). As we discussedabove, if A is I , or X , or X , or ( X X ) n X , or ( X X ) n X , for n ∈ N , then itis self-invertible in A . i.e., A − = A in A . If A is either ( X X ) n , or ( X X ) n , for any n ∈ N , then, by (3.4.3), there exists r A ∈ C , such that A − = r A − A in A ... . The above proposition characterizes the invertibility of M (2) “in A . ” ATRICIAL REPRESENTATIONS OF Γ N Trace of Certain Matrices of A . Throughout this section, we will use thesame notations used before. Let M n ( C ) be the ( n × n )-matricial algebra, for n ∈ N . Then the trace tr on M n ( C ) is well-defined as a linear functional on M n ( C ) by tr a a · · · a n a a · · · a n ... ... . . . ... a n a n · · · a nn = P nj =1 a jj . For instance, if n = 2 , then tr (cid:18) a bc d (cid:19) = a + d. Now, let A be the group algebra of Γ induced by (cid:0) C , α (cid:1) in M ( C ) , in thesense of (3.4.1) . Since the algebra A is a subalgebra of M ( C ) , one can natu-rally restrict the trace tr on M ( C ) to that on A . i.e., the pair ( A , tr ) forms a noncommutative free probability space in the sense of [8] and [9].In this section, we are interested in trace of certain types of matrices in A . Let X and X be the generating matrices of M (2) , and hence, those of A . Define anew element T of A by(3.5.1) T = X + X ∈ A . By the self-invertibility of X and X , this matrix is understood as the radialoperator (e.g., [5]) of A . Observe that T = ( X + X ) = X + ( X X + X X ) + X = I + f ( ε ) I + I = ( f ( ε ) + 2) I , and T = ( X + X ) = ( X + X ) ( X + X )= ( f ( ε ) + 2) T, and T = T T = ( f ( ε ) + 2) T = ( f ( ε ) + 2) I , and T = T T = ( f ( ε ) + 2) T, and T = T T = ( f ( ε ) + 2) T = ( f ( ε ) + 2) I , etc, where f n are in the sense of (3.2.12), f n ( z ) = P [ n ] k =0 ( − k (cid:0) nk (cid:1) (cid:18) n − k − k − (cid:19) z n − k , for all n ∈ N , and hence, f ( ε ) = ε . Inductively, one obtains that:
Theorem 3.10.
Let T = X + X be the radial operator of A , where X and X are the generating matrices of A . Then(3.5.2) T n = ( ε + 2) n I , and T n +1 = ( ε + 2) n T, in A , for all n ∈ N , where ε = det( X ) + det( X ) − det( X + X ) . Proof.
The proof of (3.5.2) is done inductively by the observations in the very aboveparagraphs. Indeed, one can get that: T n = ( f ( ε ) + 2) n I , and T n +1 = ( f ( ε ) + 2) n T, in A , for all n ∈ N . And, since f ( z ) = z, we obtain f ( ε ) = ε , where ε is in the sense of (3.2.5).By (3.5.2), we obtain the following data. Corollary 3.11.
Let T = X + X be the radial operator of A . Then(3.5.3) tr ( T n ) = (cid:26) ε + 2) n/ if n is even if n is oddfor all n ∈ N . Proof.
Let’s set(3.5.4) X j = t j s j − t j s j − t j ! in A , for j = 1 , , where ( t , s ) and ( t , s ) are strongly distinct pair in C × × C × . By (3.5.2), if T is given as above in A , then T n = ( ε + 2) n I , and T n +1 = ( ε + 2) n T, for all n ∈ N . Thus, it is not difficult to check that tr (cid:0) T n (cid:1) = tr (cid:18)(cid:18) ( ε + 2) n
00 ( ε + 2) n (cid:19)(cid:19) = ( ε + 2) n + ( ε + 2) n = 2 ( ε + 2) n , for all n ∈ N , i.e.,(3.5.5) tr ( T n ) = 2 ( ε + 2) n , for all n ∈ N . By the direct computation and by (3.5.4), one has(3.5.6) T = t + t s + s − t s + − t s − ( t + t ) ATRICIAL REPRESENTATIONS OF Γ N in A . So, one can get that: tr ( T ) = ( t + t ) + ( − ( t + t )) = 0 , by (3.5.6).Also, again by (3.5.2), we have tr ( T n +1 ) = tr (( ε + 2) n T )= ( ε + 2) n tr ( T ) = 0 , for all n ∈ N . i.e., tr ( T ) = 0 = tr (cid:0) T n +1 (cid:1) , for all n ∈ N . In summary, we have(3.5.7) tr (cid:0) T n − (cid:1) = 0 , for all n ∈ N . Therefore, by (3.5.5) and (3.5.7), we obtain the formula (3.5.3).4.
Application: A and Lucas Numbers In this section, we will use the same notations used in Section 3 above. Forinstance, let A = C (cid:2) α (Γ ) (cid:3) = C (cid:2) M (2) (cid:3) is the group algebra of Γ induced by our 2-dimensional representation (cid:0) C , α (cid:1) in M ( C ) . Here, as application of Sections 2 and 3, we study connections between analyticdata obtained from elements of A and Lucas numbers .For more details about Lucas numbers, see [7], [8], [9], [10], [11] and cited paperstherein.From the main result (3.2.23) of Section 3, we find a functional sequence ( f n ) ∞ n =1 ,with its n -th entries(4.1) f n ( z ) = P [ n ] k =0 ( − k (cid:0) nk (cid:1) (cid:18) n − k − k − (cid:19) z n − k .for all n ∈ N , satisfying( X X ) n + (( X X ) n ) − = f n ( ε ) I in A , where ε = det( X ) + det( X ) − det ( X + X ) . For any arbitrarily fixed n ∈ N , consider the (non-alternative parts of) summands(4.2) (cid:0) nk (cid:1) (cid:18) n − k − k − (cid:19) , for k = 0 , , ..., [ n ] , of f n ( z ) in (4.1).Define such quantities of (4.2) by a form of a function, g : N × N → C by(4.3) g ( n, k ) def = (cid:0) nk (cid:1) (cid:18) n − k − k − (cid:19) if k = 0 , , ..., [ n ]0 otherwise,for all ( n, k ) ∈ N × N , where N def = N ∪ { } . Note that the definition (4.3) of the function g represents the (non-alternatingparts) of summands of ( f n ) ∞ n =1 . The authors are not sure if the following theorem is already proven or not. Theycould not find the proofs in their references. So, we provide the following proof ofthe theorem.
Theorem 4.1.
Let g be a function from N × N into C be in the sense of (4.3),Then(4.4) g (1 ,
0) = 1 , and g ( n + 1 , k ) = g ( n, k ) + g ( n − , k − , for all ( n, k ) ∈ ( N \ × N .Proof. Observe first that:(4.5) g ( n, k ) = nk (cid:18) n − k − k − (cid:19) = nk ( n − k − k − n − k − − k +1)! = n ( n − k − n − k )! k ! , whenever g ( n, k ) is non-zero, for ( n, k ) ∈ N × N . By (4.5), we obtain that(4.6) g (1 ,
0) = − − − = 1 ,g (2 ,
0) = − − − = 1 , and g (2 ,
1) = − − − = 2 . Consider now that: g ( n, k ) + g ( n − , k − nk (cid:18) n − k − k − (cid:19) + n − k − (cid:18) n − − ( k − − k − − (cid:19) = n ( n − k − n − k )! k ! + ( n − n − k − n − k +1)!( k − N by (4.5) = n ( n − k +1)( n − k − k ( n − n − k − n − kk +1)! k ! = ( n + nk + n − k )( n − k − n − k +1)! k ! = ( n +1)( n − k )! k (2 n − k +1)( k − = n +1 k (cid:18) ( n + 1) − k − k − (cid:19) , and hence,(4.7) g ( n, k ) + g ( n − , k −
1) = n +1 k (cid:18) ( n + 1) − k − k − (cid:19) = g ( n + 1 , k ) . Therefore, one can get the recurrence relation (4.4), by (4.6) and (4.7).By (4.4), we can realize that the family { g ( n, k ) : ( n, k ) ∈ N × N } satisfies the following diagram(4.8) 211 21 31 4 21 5 51 6 9 21 7 14 71 8 20 16 21 9 27 30 91 10 35 50 25 2... ...where the rows of (4.8) represents n ∈ N , and the columns of (4.8) represents k ∈ N of the quantities g ( n, k ) . Observation
The above diagram (4.8) is nothing but the
Lucas triangle (e.g.,[11]) induced by the
Lucas numbers (e.g., [7], [8], [9] and [10]). (cid:3)
By using the above family { g ( n, k ) } ( n, k ) ∈ N × N of quantities obtained from (4.3),one can re-write the n -th entries f n of (4.1) as follows;(4.9) f n ( z ) = P [ n ] k =0 ( − k ( g ( n, k )) z n − k , for all n ∈ N . The new expression (4.9) shows that the coefficients of f n is determined by theLucas numbers alternatively, by (4.8). Corollary 4.2.
The coefficients of the functions f n of (4.1) are determined by theLucas numbers in the Lucas triangle (4.8) alternatively. (cid:3) Moreover, we can verify that f n +1 ( z ) = P [ n +12 ] k =0 ( − k ( g ( n + 1 , k )) z n +1 − k = P [ n +12 ] k =0 ( − k ( g ( n, k ) + g ( n − , k − z n +1 − k = P [ n +12 ] k =0 ( − k ( g ( n, k )) z n +1 − k + P [ n +12 ] k =1 ( − k ( g ( n − , k − z n − k = z (cid:16)P [ n ] k =0 ( − k ( g ( n, k − z n − k (cid:17) − P [ n − ] k =0 ( − k ( g ( n − , k − z n − − k = zf n ( z ) − f n − ( z ) . i.e., from the recurrence relation (4.7), we can re-prove the recurrence relation(3.2.16). Corollary 4.3.
Let f n ( z ) = P [ n ] k =0 ( − k ( g ( n, k )) z n − k be in the sense of (4.9), where { g ( n, k ) } ( n,k ) ∈ N × N are in the sense of (4.3).Then f ( z ) = z and f ( z ) = z − , and f n +1 ( z ) = zf n ( z ) − f n − ( z ) , for all n ≥ in N . (cid:3) Again, the above corollary demonstrates the connection between our Hilbert-space representations of Γ and Lucas numbers. References [1] I. Cho, Free Probability on Hecke Algebras and Certain Group C ∗ -AlgebrasInduced by Hecke Algebras, Opuscula Math., (2015) To Appear.[2] I. Cho, Representations and Corresponding Operators Induced by Hecke Algebras,DOI: 10.1007/s11785-014-0418-7, (2014) To Appear.[3] R. Speicher, Combinatorial Theory of the Free Product with Amalgamation andOperator-Valued Free Probability Theory, Momoir, Ame. Math. Soc., 627, (1998) ATRICIAL REPRESENTATIONS OF Γ N [4] D. Voiculescu, K. J. Dykemma, and A. Nica, Free Random Variables, CRMMonograph Ser., vol.1, ISBN-13: 978-0821811405, (1992)[5] I. Cho, and T. Gillespie, Free Probability on Hecke Algebras, DOI: 10.1007/s11785-014-0403-1, (2014) To Appear.[6] I. Cho, The Moments of Certain Perturbed Operators of the Radial Operator inthe Free Group Factors L ( F N ) , J. Anal. Appl., vol. 5, no.3, (2007) 137 - 165.[7] M. Asci, B. Cekim, and D. Tasci, Generating Matrices for Fibonaci, Lucas andSpectral Orthogonal Polynomials with Algorithms, Internat. J. Pure Appl. Math., 34,no. 2, (2007) 267 - 278.[8] M. Cetin, M. Sezer, and C. S¨uler, Lucas Polynomial Approach for System of High-Order Linear Differential Equations and Residual Error Estimation, Math. ProblemsEng., Article ID: 625984, (2015)[9] A. Nalli, and P. Haukkanem, On Generalized Fibonacci and Lucas Polynomials,Chaos, Solitons & Fractals, 42, (2009) 3179 - 3186.[10] J. Wang, Some New Results for the ( p, q ) Student Author
Ryan Golden is a senior at St. Ambrose University double majoring in Mathematicsand Biology with minors in Computer Science and Chemistry. He plans to pursue a PhD inApplied Mathematics, specifically Mathematical and Computational Neuroscience. Whilehis main interests are in the applications of mathematics, he is passionate about theoreticalmathematics as well, and is indebted to Dr. Cho for providing the opportunity to workwith and learn from him throughout his undergraduate career.
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In this paper, the authors studied matricial representationsof certain finitely presented groups generated by N -generators of order-2. As anapplication, they considered corresponding group algebras under representation.Specifically, they characterized the inverses of all group elements in terms of ma-trices in the group algebra. From the study of this characterization, the authorsrealized there are close relations between the trace of the radial operator of thealgebras and the Lucas numbers appearing in the Lucas triangle. Saint Ambrose Univ., Dept. of Math. & Stat., 518 W. Locust St., Davenport, Iowa,52803, U. S. A. /
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