Primitive orthogonal idempotents of Brandt semigroup algebras
aa r X i v : . [ m a t h . R A ] A ug PRIMITIVE ORTHOGONAL IDEMPOTENTS OF BRANDT SEMIGROUPALGEBRAS AND BRANDT SEMIGROUP CODES
YI ZHANG, JIAN-RONG LI, XIAO-SONG PENG, YAN-FENG LUO ∗ Abstract.
In this paper, we construct a complete set of primitive orthogonal idempotents forany finite Brandt semigroup algebra. As applications, we define a new class of codes calledBrandt semigroup codes and compute the Cartan matrices of some Brandt semigroup algebras.We also study the supports, Hamming distances, and minimum weights of Brandt semigroupcodes.
Key words : Brandt semigroup algebras; primitive orthogonal idempotents; coding theory : 20M25; 11T71 Introduction
In the past few decades, semigroup algebras are studied intensively, see for example, [Put88],[Okn91], [Bro00], [Ren01], [JO06], [Ste06], [Sal07], [Sch08], [Ste08], [GX09], [BBBS11], [DHST11],[MS12], [EL15].Semigroup algebras are associative algebras. Looking for a complete system of primitive or-thogonal idempotents is an important problem in the representation theory of associative alge-bras. If { e i } i ∈ I is a complete system of primitive orthogonal idempotents of a finite dimensionalalgebra A , then A ∼ = M i Ae i , (1.1)where each Ae i is a indecomposable projective module. They are also used to explicitly computethe quiver, the Cartan matrix, and the Wedderburn decomposition of the algebra, see [Bre11].Berg, Bergeron, Bhargava and Saliola [BBBS11] found a complete system of primitive or-thogonal idempotents of any R -trivial monoid algebra. Denton, Hivert, Schiling and Thiery[DHST11] gave a construction of a system of primitive orthogonal idempotents for any J -trivialmonoid algebra.Steinberg [Ste06], [Ste08] studied the arbitrary finite inverse semigroups S and computed theprimitive central idempotent associated to an irreducible representation of S . Let S be a finiteinverse semigroup, where E ( S ) is the semilattice of idempotents of S and the maximal subgroupat e is denoted by H e for all e ∈ E ( S ). Suppose that K is a field such that char( K ) ∤ | H e | . Let µ be the M¨obius function of S and χ an irreducible character of S coming from a D -class D . ∗ Corresponding author. ∗ Then the primitive central idempotents corresponding to χ are given by e χ = X e ∈ E ( D ) (cid:18) χ ( e ) | H e | X s ∈ H e χ ( s ) X t ≤ s t − µ ( t, s ) (cid:19) , (1.2)see Theorem 5.1 in [Ste06].In this paper, we construct a complete set of primitive orthogonal idempotents for any finiteBrandt semigroup algebra. Brandt semigroups are not R -trivial and in particular, they are not J -trivial. Let G denote any finite group with identity e and S = B ( G, n ) a Brandt semigroup.Suppose that G has r conjugate classes. Then a complete set of primitive orthogonal idempotentsof C S is (Theorem 3.3) (cid:26) e ij = χ i ( e ) | G | X g ∈ G χ i ( g − )( j, g, j ) | ≤ i ≤ r, ≤ j ≤ n (cid:27) . (1.3)In general, the primitive orthogonal idempotents are not primitive central idempotents, seeRemark 3.5.We apply primitive orthogonal idempotents of Brandt semigroup algebras to study the codingtheory.Abelian codes were defined as the ideals in finite abelian group algebras which were firstlyintroduced by Berman [Ber67a], [Ber67b] and MacWilliams [Mac70]. In general, a group (left)code was defined as an (left) ideal in a finite group algebra [BRC60]. Group code are studiedintensively in the past few years, see for example, [Mil79], [Aro97], [AP99], [KP01], [BR02],[BR03], [SBDR04], [FM07], [BRS08], [KAB14].A group code is called minimal if the corresponding ideal is minimal in the set of ideals of thegroup algebra. The papers [Aro97], [AP99], [FM07] computed the number of simple componentsof a semisimple finite abelian group algebra and determined all cases where this number wasminimal. This result is used to compute idempotent generators of minimal abelian codes.We define a new class of codes called Brandt semigroup codes and obtain the set of mu-tually orthogonal idempotent generators of minimal Brandt cyclic codes and minimal Brandtabelian codes. We also study the supports, Hamming distances, and minimum weights of Brandtsemigroup codes.By using complete sets of primitive orthogonal idempotents, we compute the Cartan matricesof some Brandt semigroup algebras.The paper is structured as follows. In Section 2, we recall some background information aboutsemigroup algebras, finite group characters, and group algebra codes. In Section 3, we constructa complete set of primitive orthogonal idempotents for any finite Brandt semigroup algebra andprove our main theorem (Theorem 3.3) given in Section 3 .
1. In Section 4, we give the Cartanmatrices of some Brandt semigroup algebras. In Section 5, we define and study a new class ofcodes called Brandt semigroup codes. In Section 6, we give some examples of complete sets ofprimitive orthogonal idempotents of Brandt semigroup algebras.2.
Preliminaries
In this section, we recall the definitions of semigroup algebras, Brandt semigroups, a completeset of primitive orthogonal idempotents, finite group characters, Schur orthogonality relations,Cartan matrices, and group algebra codes.
RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 3
Semigroup algebras.
Let K be a algebraically closed field, ( S, · ) be a finite semigroupwith identity element e and A be a K -algebra. The semigroup algebra of S with coefficients in A is the K -vector space AS consisting of all the formal sums P g ∈ S λ g g , where λ g ∈ A , with themultiplication defined by the formula( X g ∈ S λ g g ) · ( X h ∈ S µ h h ) = X f = gh ∈ S λ g µ h f . (2.1)Then AS is a K -algebra and the element e = 1 e is the identity of AS , where 1 is the identityelement of A , see [Okn91]. If A = K , then the elements g ∈ S form a basis of KS over K . Inthis paper we study the Brandt semigroup algebra C S , where C is the field of complex numbers.2.2. Brandt semigroups.
Let G be a group with identity element e , and I , Λ be nonemptysets. Let P = ( p λi ) be a Λ × I matrix with entries in the 0-group G = G ∪ { } , and supposethat P is regular, in the sense that no row or column of P consists entirely of zeros. Formally,( ∀ i ∈ I ) ( ∃ λ ∈ Λ) p λi = 0 , (2.2)( ∀ λ ∈ Λ) ( ∃ i ∈ I ) p λi = 0 . (2.3)Let S = ( I × G × Λ) ∪ { } , and define a multiplication on S by( i, a, λ )( j, b, µ ) = ( ( i, ap λj b, µ ) , if p λi = 0 , , if p λj = 0 , (2.4)( i, a, λ )0 = 0( i, a, λ ) = 00 = 0 . (2.5)Then S is called a completely 0-simple semigroup, see [How95].If P = E , is a identity matrix, whose diagonal elements are identity of the group G and I = Λ.Then the semigroup S is called a Brandt semigroup, denoted by B ( G, n ), where n = | I | .2.3. A complete set of primitive orthogonal idempotents.
Let A be a K -algebra withan identity (denoted by ), see I.4 in [ASS06]. A set of nonzero elements { e i } i ∈ I of A is called acomplete set of primitive orthogonal idempotents for A , if it satifies the following four properties:(i) every element e i is idempotent, namely, e i = e i for all i ∈ I ;(ii) each of the two elements are orthogonal: e i e j = e j e i = 0 for all i, j ∈ I with i = j ;(iii) every element e i is primitive: e i cannot be written as a sum, that is, if e i = x + y , then x = 0 or y = 0, where x and y are orthogonal idempotents in A ;(iv) the set { e i } i ∈ I is complete: P i ∈ I e i = . Remark 2.1 (Remark 3.2, [BBBS11]) . If { e i } i ∈ I is a maximal set of nonzero elements satisfyingconditions (i) and (ii), then { e i } i ∈ I is a complete system of primitive orthogonal idempotents(that is, (iii) and (iv) also hold). Finite group characters.
Let V be a finite dimensional vector space over a field K and ρ : G → GL ( V ) a representation of a group G on V . The character of ρ is the function χ ρ : G → K given by χ ρ ( g ) = Tr( ρ ( g )) , (2.6)where Tr is the trace. YI ZHANG, JIAN-RONG LI, XIAO-SONG PENG, YAN-FENG LUO ∗ A character χ ρ is called irreducible if ρ is an irreducible representation. The number ofconjugacy classes of G is equal to the number of irreducible characters of G and equals thenumber of isomorphism classes of irreducible KG -modules. The degree of the character χ is thedimension of ρ and this is equal to the value χ ( e ), see [Isa76].2.5. Schur orthogonality relations.
Schur orthogonality relations, see [Isa76], express a cen-tral fact about representations of finite groups. The space of complex valued class functions ofa finite group G has a natural inner product h α, β i = 1 | G | X g ∈ G α ( g ) β ( g ) , (2.7)where β ( g ) is the complex conjugate of the value of β on g .With respect to this inner product, the irreducible characters form an orthogonal basis for thespace of class functions, and this yields the orthogonality relation for the rows of the charactertable h χ i , χ j i = ( , if i = j, , if i = j. (2.8)For g, h ∈ G , the orthogonality relation for columns is as follows X χ i χ i ( g ) χ i ( h ) = ( | C G ( g ) | , if g, h are conjugate , , otherwise , (2.9)where | C G ( g ) | denotes the cardinality of the centralizer of g .2.6. Cartan Matrices.
Let A be a finite dimensional K -algebra with a complete set { e , . . . , e n } of primitive orthogonal idempotents. The Cartan matrix of A is the n × n matrix, see [ASS06], C A = c · · · c n ... . . . ... c n · · · c nn n × n , (2.10)where c ji = dim K Hom A ( Ae j , Ae i ) = dim K e i Ae j is called the Cartan invariants of A , for i, j =1 , . . . , n .By the definition, we can see that the Cartan matrix is defined with respect to a given completeset { e , e , . . . , e n } of primitive orthogonal idempotents of A . We know that e i Ae j ∼ = Hom A ( P ( j ) , P ( i )) ∼ = Hom A ( I ( j ) , I ( i )) . (2.11)Then the Cartan matrix of A records the number of homomorphisms between the indecom-posable projective A -modules and the number of homomorphisms between the indecomposableinjective A -modules. RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 5
Group algebra codes.
Let G be a finite group, and let K be an arbitrary field such thatchar( K ) ∤ | G | or char( K ) = 0. The group algebra KG of the group G over the field K is definedas the algebra over the field K consisting of all possible linear combinations α = X g ∈ G α g g, where α g ∈ K. (2.12)Any ideal V of the algebra KG is called a G -code. If the group G is abelian, the G -code V isabelian. Cyclic ( n, k )-codes over the field K are in one to one correspondence with the ideals ofthe factor ring A = K [ x ] /J, (2.13)where J = ( x n −
1) is the principal ideal, see [Ber67b].Let α = P g ∈ G α g g , β = P g ∈ G β g g . Then α = β if and only if α g = β g , (2.14)for all g ∈ G .The support of an element α ∈ KG is the set of elements of G effectively appearing in α supp( α ) = (cid:8) g ∈ G | α g = 0 (cid:9) . (2.15)Given α = P g ∈ G α g g ∈ KG , the number of elements in its support is called the weight of α ,namely ω ( α ) = | (cid:8) g | α g = 0 (cid:9) | . (2.16)3. A Complete Set of Primitive Orthogonal Idempotents for B ( G, n )In this section, we compute a complete set of the primitive orthogonal idempotents of anyfinite Brandt semigroup algebra.3.1.
Constructing primitive orthogonal idempotents.
Suppose that G is any finite groupwith identity e and has r conjugacy classes. Let S = B ( G, n ) be a finite Brandt semigroup and χ i an irreducible complex character of G . For 1 ≤ i ≤ r , 1 ≤ j ≤ n , we define e ij = χ i ( e ) | G | X g ∈ G χ i ( g − )( j, g, j ) . (3.1)We note that the element P ni =1 ( i, e, i ) is the identity of Brandt semigroup algebra C S , denotedby . Lemma 3.1 (Corollary 2.7, [Isa76]) . Let G be a finite group with the identity e and Irr( G ) the set of all irreducible complex character of G . Then | Irr( G ) | equals the number of conjugacyclasses of G and X χ ∈ Irr( G ) χ ( e ) = | G | . (3.2)By Schur orthogonality relations, we have the following lemma. YI ZHANG, JIAN-RONG LI, XIAO-SONG PENG, YAN-FENG LUO ∗ Lemma 3.2 (Theorem 2.13, [Isa76]) . Let G be a finite group with the identity e and χ i anirreducible complex character of G . Then the following holds for every h ∈ G , | G | X g ∈ G χ i ( gh ) χ j ( g − ) = δ ij χ i ( h ) χ i ( e ) , (3.3) where δ ij = ( if i = j, if i = j. (3.4)Our first main result is the following theorem. Theorem 3.3.
Let G denote any finite group with identity e and S = B ( G, n ) a Brandt semi-group. Suppose that G has r conjugacy classes. Then the elements e ij = χ i ( e ) | G | X g ∈ G χ i ( g − )( j, g, j ) , (3.5) where ≤ i ≤ r , ≤ j ≤ n , form a complete set of primitive orthogonal idempotents of theBrandt semigroup algebra C S .Proof. Firstly, we prove the set { e ij } ≤ i ≤ r, ≤ j ≤ n is complete.Since G has r conjugacy classes, G has r irreducible representations. For any 1 ≤ j ≤ n , wehave r X i =1 e ij = 1 | G | r X i =1 X g ∈ G χ i ( e ) χ i ( g − )( j, g, j )= 1 | G | r X i =1 χ i ( e ) (cid:0) χ i ( e − )( j, e, j ) + X g ∈ G \{ e } χ i ( g − )( j, g, j ) (cid:1) = 1 | G | r X i =1 χ i ( e ) χ i ( e − )( j, e, j ) + 1 | G | X g ∈ G \{ e } r X i =1 χ i ( e ) χ i ( g − )( j, g, j ) . Obviously, e and g ∈ G \{ e } are not conjugate. According to Schur orthogonality relations (2.9),we have r X i =1 χ i ( e ) χ i ( g ) = 0 , where g = e .Hence r X i =1 χ i ( e ) χ i ( g − ) = 0 . Thus r X i =1 e ij = 1 | G | r X i =1 χ i ( e ) χ i ( e − )( j, e, j ) = 1 | G | r X i =1 ( χ i ( e )) ( j, e, j ) . RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 7
By Lemma 3.1, we have r X i =1 ( χ i ( e )) = | G | . Then r X i =1 e ij = ( j, e, j ) . Thus n X j =1 r X i =1 e ij = n X j =1 ( j, e, j ) = . Secondly, we prove that the elements of { e ij } ≤ i ≤ r, ≤ j ≤ n are pairwise orthogonal. For any j = j , 1 ≤ j , j ≤ n , 1 ≤ i , i ≤ r , e i j e i j = χ i ( e ) | G | X g ′ ∈ G χ i ( g ′− )( j , g ′ , j ) χ i ( e ) | G | X g ∈ G χ i ( g − )( j , g, j ) (3.6)= 0 . (3.7)For any j = j , i = i , e i j e i j = χ i ( e ) | G | X g ′ ∈ G χ i ( g ′− )( j , g ′ , j ) χ i ( e ) | G | X g ∈ G χ i ( g − )( j , g, j ) (3.8)= χ i ( e ) χ i ( e ) | G | X g ′ ∈ G X g ∈ G χ i ( g ′− ) χ i ( g − )( j , g ′ g, j ) (3.9)= χ i ( e ) χ i ( e ) | G | X g ′′ ∈ G X g ∈ G χ i ( gg ′′− ) χ i ( g − )( j , g ′′ , j ) . (3.10)By Lemma 3.2, X g ∈ G χ i ( gg ′′− ) χ i ( g − ) = δ i i χ i ( g ′′− ) χ i ( e ) . (3.11)Then, for any i = i , e i j e i j = 0 . (3.12)Hence, the elements of { e ij } ≤ i ≤ r, ≤ j ≤ n are pairwise orthogonal.In the following, we prove that each element of { e ij } ≤ i ≤ r, ≤ j ≤ n is idempotent. Let e st beany element of the set { e ij } ≤ i ≤ r, ≤ j ≤ n , where 1 ≤ s ≤ r , 1 ≤ t ≤ n . Then e st = e st (cid:0) n X j =1 r X i =1 e ij (cid:1) = e st = e st . (3.13) YI ZHANG, JIAN-RONG LI, XIAO-SONG PENG, YAN-FENG LUO ∗ Finally, we prove that each element of { e ij } ≤ i ≤ r, ≤ j ≤ n is primitive. Suppose that f ∈ C S , f = f , f = 0, f / ∈ { e ij } ≤ i ≤ r, ≤ j ≤ n , and f is orthogonal to every e ij , 1 ≤ i ≤ r , 1 ≤ j ≤ n .Then f = f = f (cid:0) n X j =1 r X i =1 e ij (cid:1) = 0 . (3.14)This contradicts the assumption that f = 0. Therefore { e ij } ≤ i ≤ r, ≤ j ≤ n is a maximal set ofnonzero elements satisfying conditions (i) and (ii) in Section 2.3. By Remark 2.1, each elementof { e ij } ≤ i ≤ r, ≤ j ≤ n is primitive. (cid:3) Corollary 3.4.
Let G = h a | a k = e i be a finite cyclic group and S a Brandt semigroup with | Λ | = | I | = n . Then for ≤ p ≤ k , ≤ q ≤ n − , the elements e p + qk ( n +1) = 1 k k X j =1 ω − jp ( q + 1 , a j , q + 1) , (3.15) where ω = e πik , i = √− , form a complete set of primitive orthogonal idempotents of the Brandtsemigroup algebra C S .Proof. This follows directly from Theorem 3.3. (cid:3)
Remark 3.5.
Let S = B ( G, n ) be a finite Brandt semigroup. If n = 1 , the primitive orthogonalidempotents of C S are precisely primitive central idempotents. If n ≥ , in general, the primitiveorthogonal idempotents are not primitive central idempotents. We can see the following example.Let G = { e } , I = { , } . Then we have the Brandt semigroup S = B = h a, b | a = b = 0 , aba = a, bab = b i . We also have the identity = X i =1 ( i, e, i ) . By Theorem 3.4, the elements e = (1 , e, , e = (2 , e, form a complete set of primitive orthogonal idempotents in the semigroup algebra C S .Let C S = { P i =1 k i s i | k i ∈ C , s i ∈ S } . Then we have e C S = (cid:8) k (1 , e,
1) + k (1 , e, | k , k ∈ C (cid:9) , C Se = (cid:8) k (1 , e,
1) + k (2 , e, | k , k ∈ C (cid:9) . Obviously, e C S = C Se . Then the primitive orthogonal idempotents are not primitive central idempotents in the semigroupalgebra C S . RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 9 The Cartan Matrices of some Brandt semigroup algebras
In this section, we compute the Cartan matrices of some Brandt semigroup algebras.We have the following theorems.
Theorem 4.1.
The Cartan matrix of the Brandt semigroup algebra C B ( { e } , n ) is C C B ( { e } ,n ) = · · ·
11 1 · · · ... ... . . . ... · · · n × n . (4.1) Proof.
By Corollary 3.4, a complete set of primitive orthogonal idempotents of Brandt semigroupalgebra C B ( { e } , n ) is (cid:8) e i = ( i, e, i ) | ≤ i ≤ n (cid:9) . For any 1 ≤ λ, µ ≤ n , we have C B ( { e } , n ) = (cid:26) n X µ =1 n X λ =1 k λµ ( λ, e, µ ) | k λµ ∈ C (cid:27) . Hence C B ( { e } , n ) e = (cid:26) n X µ =1 n X λ =1 k λµ ( λ, e, µ )(1 , e, | k λµ ∈ C (cid:27) = (cid:26) X λ =1 k λ ( λ, e, | k λ ∈ C (cid:27) . Therefore, for any 1 ≤ i ≤ n , e i C B ( { e } , n ) e = (cid:26) k i ( i, e, | k i ∈ C (cid:27) . Thus dim C e i C B ( { e } , n ) e = 1 . By a similar argument, for any 1 ≤ i, j ≤ n ,dim C e i C B ( { e } , n ) e j = 1 . (cid:3) Theorem 4.2.
Let G = h a | a = e i be a cyclic group. Then the Cartan matrix of the Brandtsemigroup algebra C B ( G , n ) is C C B ( G ,n ) = E E · · ·
EE E · · · E ... ... . . . ... E E · · · E n × n , (4.2) where E is the identity matrix of order . ∗ Proof.
By Corollary 3.4, a complete set of primitive orthogonal idempotents of Brandt semigroupalgebra C B ( G , n ) is (cid:8) e i − , e i | ≤ i ≤ n (cid:9) , where e i − = 12 ( i, e, i ) −
12 ( i, a, i ) , e i = 12 ( i, e, i ) + 12 ( i, a, i ) . (4.3)For any 1 ≤ λ, µ ≤ n , we have C B ( G , n ) = (cid:26) n X λ =1 n X µ =1 2 X j =1 k λjµ ( λ, a j , µ ) | k λjµ ∈ C (cid:27) . Hence C B ( G , n ) e = (cid:26) n X λ =1 2 X j =1 k λj ( λ, a j , (cid:0) (1 , e, − (1 , a, (cid:1) | k λj ∈ C (cid:27) (4.4)= (cid:26) n X λ =1 ( k λ − k λ ) (cid:0) ( λ, e, − ( λ, a, (cid:1) | k λ , k λ ∈ C (cid:27) . (4.5)Therefore, for any 1 ≤ i ≤ n , e i C B ( G , n ) e = 0 , (4.6) e i − C B ( G , n ) e = (cid:26)
12 ( k i − k i ) (cid:0) ( i, e, − ( i, a, (cid:1) | k i , k i ∈ C (cid:27) . (4.7)Thus dim C e i − C B ( G , n ) e = 1 , dim C e i C B ( G , n ) e = 0 . (4.8)Similarly, for any 1 ≤ i, j ≤ n ,dim C e i − C B ( G , n ) e j − = 1 , dim C e i C B ( G , n ) e j − = 0 . (4.9)By a similar argument, for any 1 ≤ i, j ≤ n ,dim C e i − C B ( G , n ) e j = 0 , dim C e i C B ( G , n ) e j = 1 . (4.10) (cid:3) Theorem 4.3.
Let G = h a | a k = e, k ≥ i be a cyclic group. Then the Cartan matrix of theBrandt semigroup algebra C B ( G, n ) is C C B ( G,n ) = kE kE · · · kEkE kE · · · kE ... ... . . . ... kE kE · · · kE kn × kn , (4.11) where E is the identity matrix of order k . RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 11
Proof.
By Corollary 3.4, a complete set of primitive orthogonal idempotents of Brandt semigroupalgebra C B ( G, n ) is (cid:26) k k X j =1 ω − jp ( q + 1 , a j , q + 1) | ≤ p ≤ k, ≤ q ≤ n − (cid:27) . (4.12)We list all them by the following matrix: e = k P kj =1 ω − j (1 , a j , e k +1 = k P kj =1 ω − j (2 , a j , · · · e ( n − k +1 = k P kj =1 ω − j ( n, a j , n ) e = k P kj =1 ω − j (1 , a j , e k +2 = k P kj =1 ω − j (2 , a j , · · · e ( n − k +2 = k P kj =1 ω − j ( n, a j , n )... ... . . . ... e k = k P kj =1 ω − kj (1 , a j , e k = k P kj =1 ω − kj (2 , a j , · · · e nk = k P kj =1 ω − kj ( n, a j , n ) n × k . For any 1 ≤ λ, µ ≤ n , we have C B ( G, n ) = (cid:26) n X λ =1 n X µ =1 k X j =1 k λjµ ( λ, a j , µ ) | k λjµ ∈ C (cid:27) . Hence C B ( G, n ) e = (cid:26) k n X λ =1 k X j =1 k λj ( λ, a j , k X j ′ =1 ω − j ′ (1 , a j ′ , | k λj ∈ C (cid:27) = (cid:26) k n X λ =1 (cid:0) ( k λ + k λ ω − ( k − + k λ ω − ( k − + · · · + k λk ω − )( λ, a, k λ ω − + k λ + k λ ω − ( k − + · · · + k λk ω − )( λ, a , · · · + ( k λ ω − ( k − + k λ ω − ( k − + · · · + k λk )( λ, e, (cid:1) | k λj ∈ C (cid:27) . ∗ Therefore e k +1 C B ( G, n ) e = (cid:26) k (cid:18)(cid:0) ω − ( k ω − ( k − + k ω − ( k − + · · · + k k )+ ω − ( k ω − ( k − + k ω − ( k − + · · · + k k ω − ( k − )+ · · · + ω − k ( k + k ω − ( k − + · · · + k k ω − ) (cid:1) (2 , a, (cid:0) ω − ( k + k ω − ( k − + · · · + k k ω − )+ ω − ( k ω − ( k − + k ω − ( k − + · · · + k k )+ · · · + ω − k ( k ω − + k + · · · + k k ω − ) (cid:1) (2 , a , · · · + (cid:0) ω − ( k ω − ( k − + k ω − ( k − + · · · + k k ω − ( k − )+ ω − ( k ω − ( k − + k ω − ( k − + · · · + k k ω − ( k − )+ · · · + ω − k ( k ω − ( k − + k ω − ( k − + · · · + k k ) (cid:1) (2 , e, (cid:19) | k j ∈ C (cid:27) = (cid:26) k (cid:18) ( k + k ω − ( k − + k ω − ( k − + · · · + k k ω − )(2 , a, k ω − + k + k ω − ( k − + · · · + k k ω − )(2 , a , · · · + ( k ω − ( k − + k ω − ( k − + · · · + k k )(2 , e, | k j ∈ C (cid:19)(cid:27) ,e C B ( G, n ) e = (cid:26) k (1 + ω − + ω − + · · · + ω − ( k − ) k X j =1 k j (1 , a j , | k j ∈ C (cid:27) = 0 . Similarly, for any 0 ≤ q ≤ n − e qk +1 C B ( G, n ) e = (cid:26) k (cid:18) ( k ( q +1)11 + k ( q +1)21 ω − ( k − + k ( q +1)31 ω − ( k − + · · · + k k ω − )( q + 1 , a, k ( q +1)11 ω − + k ( q +1)21 + k ( q +1)31 ω − ( k − + · · · + k ( q +1) k ω − )( q + 1 , a , · · · + ( k ( q +1)11 ω − ( k − + k ( q +1)21 ω − ( k − + · · · + k ( q +1) k )( q + 1 , e, | k ( q +1) j ∈ C (cid:19)(cid:27) . For any 0 ≤ q ≤ n −
1, 2 ≤ l ≤ k , e qk + l C B ( G, n ) e = 0 . RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 13
Thus, for any k ≥ C e qk +1 C B ( G, n ) e = k, dim C e qk + l C B ( G , n ) e = 0 . The other cases are similar. (cid:3) Brandt semigroup Codes
In this section, we introduce a new class of codes called Brandt semigroup codes, and studythe supports, Hamming distances, and minimum weights of Brandt semigroup codes.We introduce a new class of codes called Brandt semigroup codes as follows.
Definition 5.1.
Let S = B ( G, n ) be a Brandt semigroup. Then a Brandt semigroup algebracode C in C S , or a Brandt semigroup code for short, is a one side ideal of C S . A code C iscalled a Brandt abelian code if the finite group G is a abelian group and the index n = { } .Otherwise, the code C is called a Brandt non-abelian code. Definition 5.2.
Let C = P ( j,g,j ) ∈ S α ( j,g,j ) ( j, g, j ) ∈ C S , C = P ( j,g,j ) ∈ S β ( j,g,j ) ( j, g, j ) ∈ C S ,where α ( j,g,j ) , β ( j,g,j ) ∈ C , be two Brandt semigroup codes. Then the number of the two elementsof the support in which the coefficients differ is called the Hamming distance of C and C ,namely, d ( C , C ) = | (cid:8) ( j, g, j ) | α ( j,g,j ) = β ( j,g,j ) , ( j, g, j ) ∈ S (cid:9) | . (5.1) For a code C of C S , we define the minimum weight of C as: ω ( C ) = min (cid:8) ω ( α ) | α ∈ C, α = 0 (cid:9) . (5.2) Theorem 5.3.
Let S = B ( G, be a finite Brandt semigroup and G = h a | a k = e i a finitecyclic group. Then (cid:26) e p = 1 k k X j =1 ω − jp (1 , a j , | ≤ p ≤ k, ω = e πik (cid:27) (5.3) is a set of mutually orthogonal idempotent generators of minimal Brandt cyclic codes of C B ( G, .Let G be a finite abelian group with identity e of conjugate classes r . Then (cid:26) e i = χ i ( e ) | G | X g ∈ G χ i ( g − )(1 , g, | ≤ i ≤ r (cid:27) (5.4) is a set of mutually orthogonal idempotent generators of minimal Brandt abelian codes of C B ( G, .Proof. This result follows from Theorem 3.3 in Section 3 and Theorem 2 in [Sab93]. (cid:3)
Lemma 5.4 (Theorem 2.5.11, [MS02]) . Let A = L ti =1 A i be a decomposition of a semisimplealgebra as a direct sum of minimal left ideals. Then, there exists a family of elements { e , . . . , e t } which consist a complete set of primitive orthogonal idempotents of A .Conversely, if there exists a family of idempotents { e , . . . , e t } satisfying the four propertiesof the Definition 2.3, then the left ideals A i = Ae i are minimal and A = L ti =1 A i . Theorem 5.5.
Let S = B ( G, n ) be a finite Brandt semigroup. Suppose that n ≥ and { e ij | ≤ i ≤ r, ≤ j ≤ n } is a complete set of primitive orthogonal idempotents of C S . Then { C Se ij | ≤ i ≤ r, ≤ j ≤ n } are minimal Brandt non-abelian codes. ∗ Proof.
By Corollary 4.5 in [Ste06], the Brandt semigroup algebra C S is semisimple. By Lemma5.4 and Definition 5.1, C Se ij , 1 ≤ i ≤ r , 1 ≤ j ≤ n , are minimal Brandt non-abelian codes. (cid:3) Proposition 5.6 (Theorem 2.5.10, [MS02]) . Let S be a finite Brandt semigroup. Then everyleft ideal L of C S is of the form L = C Se , where e ∈ C S is an idempotent. Lemma 5.7 (Proposition 4.5, [ASS06]) . Let B = A/radA , where A is a K -algebra. Then everyleft ideal I of B is a direct sum of simple right ideals of the form Be , where e is a primitiveidempotent of B . Theorem 5.8.
Let S be a finite Brandt semigroup. Then every Brandt semigroup code C of C S is a direct sum of minimal Brandt semigroup codes of the form C Se , where e is a primitiveidempotent of C S .Proof. This follows directly from Lemma 5.7. (cid:3)
We are mostly interseted in that how to get all Brandt semigroup codes by a complete set ofprimitive orthogonal idempotents of C S . Using the basic group table matrix of G , W. S. Park[Par97] found all the idempotents of the group algebra KG , where K is a algebraically closedfield of characteristic 0 and G is a cyclic group of order n . Suppose that the index set of Brandtsemigroup S is trivial, then we can get all the Brandt semigroup codes by a complete set ofprimitive orthogonal idempotents of C S . Theorem 5.9.
Let S = B ( G, be a finite Brandt semigroup and G a cyclic group of order n .Then every Brandt semigroup code C of C S is a direct sum of minimal Brandt semigroup codesof the form C Se i , where { e i } ≤ i ≤ n is a complete set of primitive orthogonal idempotents of C S .Proof. By Theorem 3.1 in [Par97], if r = 1 /n , the idempotents P n − j =0 r j (1 , g j ,
1) are preciselyform a complete set of primitive orthogonal idempotents of C S which are denoted e i , where1 ≤ i ≤ n . Then C Se i are all minimal Brandt semigroup codes. Since all the other idempotentscan be represented by e i , 1 ≤ i ≤ n , by combination. Then every Brandt semigroup code C of C S is a direct sum of some minimal Brandt semigroup codes. (cid:3) Example 5.10.
Let G = h a | a = e i be a cyclic group, Λ = I = { } , and S = B ( G , aBrandt semigroup. Then the elements e = 14 (cid:0) (1 , e,
1) + (1 , a,
1) + (1 , a ,
1) + (1 , a , (cid:1) ,e = 14 (cid:0) (1 , e, − (1 , a,
1) + (1 , a , − (1 , a , (cid:1) ,e = 14 (cid:0) (1 , e,
1) + i(1 , a, − (1 , a , − i(1 , a , (cid:1) ,e = 14 (cid:0) (1 , e, − i(1 , a, − (1 , a ,
1) + i(1 , a , (cid:1) , RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 15 form a complete set of primitive orthogonal idempotents of C S . The following codes C = C Se = (cid:26) k + k + k + k (cid:0) (1 , e,
1) + (1 , a,
1) + (1 , a ,
1) + (1 , a , (cid:1) | k , k , k , k ∈ C (cid:27) ,C = C Se = (cid:26) k − k + k − k (cid:0) (1 , e, − (1 , a,
1) + (1 , a , − (1 , a , (cid:1) | k , k , k , k ∈ C (cid:27) ,C = C Se = (cid:26) k − k i − k + k i4 (cid:0) (1 , e, − (1 , a , (cid:1) + k i + k − k i − k (cid:0) (1 , a, − (1 , a , (cid:1) | k , k , k , k ∈ C (cid:27) ,C = C Se = (cid:26) k + k i − k − k i4 (cid:0) (1 , e, − (1 , a , (cid:1) − k i − k − k i + k (cid:0) (1 , a, − (1 , a , (cid:1) | k , k , k , k ∈ C (cid:27) , are minimal Brandt cyclic codes, where C S = { P i =1 k i s i | k i ∈ C , s i ∈ S } . We list all the Brandtcyclic codes of C S and their dimensions and weights in Table 1. In Table 1, C , C , C , C , C are minimal Brandt cyclic codes and C , C , C , C , C , C , C , C , C , C , C are non-minimal Brandt cyclic codes. Brandt semigroup code Dimension Minimum weight C = 0 0 0 C = { k + k + k + k (cid:0) (1 , e,
1) + (1 , a,
1) + (1 , a ,
1) + (1 , a , (cid:1) | k , k , k , k ∈ C } C = { k − k + k − k (cid:0) (1 , e, − (1 , a,
1) + (1 , a , − (1 , a , (cid:1) | k , k , k , k ∈ C } C = { k − k i − k + k i4 (cid:0) (1 , e, − (1 , a , (cid:1) + k i+ k − k i − k (cid:0) (1 , a, − (1 , a , (cid:1) | k , k , k , k ∈ C } C = { k + k i − k − k i4 (cid:0) (1 , e, − (1 , a , (cid:1) − k i − k − k i+ k (cid:0) (1 , a, − (1 , a , (cid:1) | k , k , k , k ∈ C } C = { k + k (cid:0) (1 , e,
1) + (1 , a , (cid:1) + k + k (cid:0) (1 , a,
1) + (1 , a , (cid:1) | k , k , k , k ∈ C } C = { k − k (cid:0) (1 , e, − (1 , a , (cid:1) + k − k (cid:0) (1 , a, − (1 , a , (cid:1) | k , k , k , k ∈ C } C = { k − i) k k (1 ,e, k − i) k k (1 ,a, k − i) k k (1 ,a , k − i) k k (1 ,a , | k ,k ,k ,k ∈ C } C = { k k − i) k (1 ,e, k k − i) k (1 ,a, k k − i) k (1 ,a , k k − i) k (1 ,a , | k ,k ,k ,k ∈ C } C = { k − i) k − (1 − i) k (1 ,e, k − (1+i) k − (1 − i) k (1 ,a, k − (1+i) k − (1 − i) k (1 ,a , k − (1+i) k − (1 − i) k (1 ,a , | k ,k ,k ,k ∈ C } C = { k − (1 − i) k − (1+i) k (1 ,e, k − (1 − i) k − (1+i) k (1 ,a, k − (1 − i) k − (1+i) k (1 ,a , k − (1 − i) k − (1+i) k (1 ,a , | k ,k ,k ,k ∈ C } C = { k − i k k k (1 ,e, k − i k k k (1 ,a, k − i k k k (1 ,a , k − i k k k (1 ,a , | k ,k ,k ,k ∈ C } C = { k k k − i k (1 ,e, k k k − i k (1 ,a, k k k − i k (1 ,a , k k k − i k (1 ,a , | k ,k ,k ,k ∈ C } C = { k k − k k (1 ,e, k k − k k (1 ,a, k k − k k (1 ,a , k k − k k (1 ,a , | k ,k ,k ,k ∈ C } C = { k − k − k − k (1 ,e, k − k − k − k (1 ,a, k − k − k − k (1 ,a , k − k − k − k (1 ,a , | k ,k ,k ,k ∈ C } C = C S Table 1.
Brandt semigroup codes in C B ( G , S is not trivial, the complete set of primitive orthogonalidempotents of C S is not unique. we may not get all the idempotents by a complete set ofprimitive orthogonal idempotents of C S by combination. See the following example. ∗ Example 5.11.
Let G = h a | a = e i be a cyclic group, Λ = I = { , } , and S = B ( G , aBrandt semigroup. By Corollary 3.4, the elements e = 12 k X j =1 ω − j (1 , a j ,
1) = 12 (1 , e, −
12 (1 , a, ,e = 12 k X j =1 ω − j (1 , a j ,
1) = 12 (1 , e,
1) + 12 (1 , a, ,e = 12 k X j =1 ω − j (2 , a j ,
2) = 12 (2 , e, −
12 (2 , a, ,e = 12 k X j =1 ω − j (2 , a j ,
2) = 12 (2 , e,
2) + 12 (2 , a, , form a complete set of primitive orthogonal idempotents of C S . The following codes C = C Se = (cid:26) k − k (cid:0) (1 , e, − (1 , a, (cid:1) + k − k (cid:0) (2 , e, − (2 , a, (cid:1) | k , k , k , k ∈ C (cid:27) ,C = C Se = (cid:26) k + k (cid:0) (1 , e,
1) + (1 , a, (cid:1) + k + k (cid:0) (2 , e,
1) + (2 , a, (cid:1) | k , k , k , k ∈ C (cid:27) ,C = C Se = (cid:26) k − k (cid:0) (1 , e, − (1 , a, (cid:1) + k − k (cid:0) (2 , e, − (2 , a, (cid:1) | k , k , k , k ∈ C (cid:27) ,C = C Se = (cid:26) k + k (cid:0) (1 , e,
2) + (1 , a, (cid:1) + k + k (cid:0) (2 , e,
2) + (2 , a, (cid:1) | k , k , k , k ∈ C (cid:27) , are minimal Brandt non-abelian codes, where C S = { P i =1 k i s i | k i ∈ C , s i ∈ S } . We can easilycheck that the element e z = (1 , e,
1) + z (2 , e,
1) + z (2 , a, is a idempotent of C S , where z ∈ C .However, C Se z = ( k + k z + k z )(1 , e,
1) + ( k + k z + k z )(1 , a,
1) (5.5)+ ( k + k z + k z )(2 , e,
1) + ( k + k z + k z )(2 , a,
1) (5.6) can not be written as a direct sum of minimal Brandt semigroup codes. Examples of primitive orthogonal idempotents of Brandt semigroup algebras
In this section, we give some examples of complete sets of primitive orthogonal idempotentsof Brandt semigroup algebras.
RIMITIVE ORTHOGONAL IDEMPOTENTS AND BRANDT SEMIGROUP CODES 17
Example 1.
Let G = h a | a = e i be a cyclic group and Λ = I = { } . Then the elements e = 13 X j =1 ω − j (1 , a j ,
1) = 13 (1 , e,
1) + 13 (1 , a,
1) + 13 (1 , a , , (6.1) e = 13 X j =1 ω − j (1 , a j ,
1) = 13 (1 , e, − ( 16 − √ , a, − ( 16 + √ , a , , (6.2) e = 13 X j =1 ω − j (1 , a j ,
1) = 13 (1 , e, − ( 16 + √ , a, − ( 16 − √ , a , , (6.3) e = 13 X j =1 ω − j (2 , a j ,
2) = 13 (2 , e,
2) + 13 (2 , a,
2) + 13 (2 , a , , (6.4) e = 13 X j =1 ω − j (2 , a j ,
2) = 13 (2 , e, − ( 16 − √ , a, − ( 16 + √ , a , , (6.5) e = 13 X j =1 ω − j (2 , a j ,
2) = 13 (2 , e, − ( 16 + √ , a, − ( 16 − √ , a , , (6.6)where w = e πi = cos π + i sin π , i = √−
1, form a complete set of primitive orthogonalidempotents of C B ( G , Example 2.
Let G = h a i × h b i ∼ = C × C be an abelian group of order 4. The charactertable of G is as follows. G e a b abχ χ χ χ Table 2.
Character table of abelian group G .Let Λ = I = { } . Then the Brandt semigroup algebra C B ( G,
1) has four primitive orthogonalidempotents: e = 14 (cid:0) (1 , e,
1) + (1 , a,
1) + (1 , b,
1) + (1 , ab, (cid:1) ,e = 14 (cid:0) (1 , e, − (1 , a,
1) + (1 , b, − (1 , ab, (cid:1) ,e = 14 (cid:0) (1 , e,
1) + (1 , a, − (1 , b, − (1 , ab, (cid:1) ,e = 14 (cid:0) (1 , e, − (1 , a, − (1 , b,
1) + (1 , ab, (cid:1) . ∗ Example 3.
Let G = S . This group has 6 elements:1 , (12) , (13) , (23) | {z } , (123) , (132) | {z } , (6.7)where |{z} means the elements are conjugate. There are three conjugacy classes.Next, we have the character table of S . S χ χ χ Table 3.
Character table of S .Let Λ = I = { , } . Then the Brandt semigroup algebra C B ( S ,
2) has six primitive orthog-onal idempotents: e = 16 (cid:0) (1 , ,
1) + (1 , (123) ,
1) + (1 , (132) ,
1) + (1 , (12) ,
1) + (1 , (13) ,
1) + (1 , (23) , (cid:1) ,e = 16 (cid:0) (1 , ,
1) + (1 , (123) ,
1) + (1 , (132) , − (1 , (12) , − (1 , (13) , − (1 , (23) , (cid:1) ,e = 16 (cid:0) (1 , , − , (123) , − , (132) , (cid:1) ,e = 16 (cid:0) (2 , ,
2) + (2 , (123) ,
2) + (2 , (132) ,
2) + (2 , (12) ,
2) + (2 , (13) ,
2) + (2 , (23) , (cid:1) ,e = 16 (cid:0) (2 , ,
2) + (2 , (123) ,
2) + (2 , (132) , − (2 , (12) , − (2 , (13) , − (2 , (23) , (cid:1) ,e = 16 (cid:0) (2 , , − , (123) , − , (132) , (cid:1) . These primitive orthogonal idempotents are obtained from Theorem 3.3. For example, e = χ (1)6 (cid:0) χ (1) − (1 , ,
1) + χ (123) − (1 , (123) ,
1) + χ (132) − (1 , (132) , (cid:1) = 16 (cid:0) (1 , , − , (123) , − , (132) , (cid:1) . Acknowledgement
The authors are supported by the National Natural Science Foundation of China (no. 11371177,11501267, 11401275), and the Fundamental Research Funds for the Central Universities of China(no. lzujbky-2015-78).
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E-mail address : [email protected] Jian-Rong Li: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P. R.China.
E-mail address : [email protected] Xiao-song Peng: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China.
E-mail address : [email protected] Yan-Feng Luo: School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, P.R. China.
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