On modular representations of semigroups Sp × Tp

On modular representations of semigroups Sp × Tp

Let p be simple, and let Sp and Tp be the symmetric group and the symmetric semigroup of degree p, respectively. The theorem of this paper says hat the direct product Sp × Tp are of wild representation type over any field of characteristic p. The main case is p = 2.

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Бібліографічні деталі
Дата:2013
Автори: Bondarenko, V., Kostyshyn, E.
Формат: Стаття
Мова:English
Опубліковано: Інститут прикладної математики і механіки НАН України 2013
Назва видання:Algebra and Discrete Mathematics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/152304
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Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:On modular representations of semigroups Sp × Tp / V. Bondarenko, E. Kostyshyn // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 16–19. — Бібліогр.: 5 назв. — англ.

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spelling irk-123456789-1523042019-06-10T01:25:29Z On modular representations of semigroups Sp × Tp Bondarenko, V. Kostyshyn, E. Let p be simple, and let Sp and Tp be the symmetric group and the symmetric semigroup of degree p, respectively. The theorem of this paper says hat the direct product Sp × Tp are of wild representation type over any field of characteristic p. The main case is p = 2. 2013 Article On modular representations of semigroups Sp × Tp / V. Bondarenko, E. Kostyshyn // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 16–19. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC:16G, 20M30. http://dspace.nbuv.gov.ua/handle/123456789/152304 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description Let p be simple, and let Sp and Tp be the symmetric group and the symmetric semigroup of degree p, respectively. The theorem of this paper says hat the direct product Sp × Tp are of wild representation type over any field of characteristic p. The main case is p = 2.
format Article
author Bondarenko, V.
Kostyshyn, E.
spellingShingle Bondarenko, V.
Kostyshyn, E.
On modular representations of semigroups Sp × Tp
Algebra and Discrete Mathematics
author_facet Bondarenko, V.
Kostyshyn, E.
author_sort Bondarenko, V.
title On modular representations of semigroups Sp × Tp
title_short On modular representations of semigroups Sp × Tp
title_full On modular representations of semigroups Sp × Tp
title_fullStr On modular representations of semigroups Sp × Tp
title_full_unstemmed On modular representations of semigroups Sp × Tp
title_sort on modular representations of semigroups sp × tp
publisher Інститут прикладної математики і механіки НАН України
publishDate 2013
url http://dspace.nbuv.gov.ua/handle/123456789/152304
citation_txt On modular representations of semigroups Sp × Tp / V. Bondarenko, E. Kostyshyn // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 16–19. — Бібліогр.: 5 назв. — англ.
series Algebra and Discrete Mathematics
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fulltext Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 16 (2013). Number 1. pp. 16 – 19 © Journal “Algebra and Discrete Mathematics” On modular representations of semigroups Sp × TpSp × TpSp × Tp Vitaliy M. Bondarenko, Elina M. Kostyshyn Communicated by V. V. Kirichenko Abstract. Let p be simple, and let Sp and Tp be the sym- metric group and the symmetric semigroup of degree p, respectively. The theorem of this paper says that the direct product Sp × Tp are of wild representation type over any field of characteristic p. The main case is p = 2. Let k be a field. A semigroup is called of tame representation type (resp. of wild representation type) over k if so is the problem of classifying its representations (see precise general definitions in [1]). We give the precise definition of semigroups of wild representation type in matrix language. For a semigroup S and a k-algebra Λ, we denote by RΛ(S) the set of all matrix representations of S over Λ; Rk(Λ) denotes the category of matrix representations of Λ over k. A semigroup S is called of wild representation type (or simply wild) over k if there exists a matrix representation M of S over Λ = K2 = k < x, y > such that the following conditions hold: 1) the matrix representation M ⊗ X (of S over k) with X ∈ Rk(Λ) is indecomposable if so is X; 2) the matrix representations M ⊗ X and M ⊗ X ′ are nonequivalent if so are X and X ′. 2010 MSC: 16G, 20M30. Key words and phrases: matrix, wild, transformation, symmetric semigroup, modular representations. V. Bondarenko, E. Kostyshyn 17 Here K2 = k < x, y > denotes the free associative k-algebra in two noncommuting variables x and y. We call such an M a perfect representation of S over Λ. In practice, to simplify the proofs of wildness (not only semigroup but also other objects) one can replace K2 by any wild k-algebra. The main result of this paper is the following theorem. Theorem. Let k be a field of characteristic p 6= 0 and let Sp and Tp be the symmetric group and the symmetric semigroup of degree p, respectively. Then the semigroup Sp × Tp is wild over k. Here × denotes, as usual, the sign of the direct product. Note that Tp and Sp × Tp are monoids. Since the factor semigroup of Tp by its only maximal two-sided ideal (generated by all the non-invertible elements) is isomorphic to Sp, the semigroup Sp × Tp is wild for p 6= 2 by the criterion of tameness and wildness of finite groups [2]. In case p = 2 we will indicate a perfect representation of Sp × Tp over the k-algebra Λ = kΓ of paths of the quiver Γ with two vertices p1, p2 and two arrows x : p1 → p1, y : p1 → p2 (this quiver is wild [3, 4]). The monoid T2 of transformations of the set {1, 2} is generated by the elements a, b, where a(1) = 2, a(2) = 1, b(1) = 2, b(2) = 2, with defining relations a2 = 1, b2 = b, ab = b [5]. Obviously that the monoid S2 × T2 is generated by the elements g, a, b with the additional relations g2 = 1, ga = ag, gb = bg (g denotes the non-identity element of S2). Consider the next matrix representation γ of S2 × T2 over the algebra Λ = kΓ: γ(g) =      1 0 0 0 0 1 0 0 0 y 1 x 0 0 0 1      , γ(a) =      1 1 0 0 0 1 0 0 0 0 1 1 0 0 0 1      , γ(b) =      1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0      (γ(1) is equal to the identity matrix). We will prove that γ is a perfect representation. Let ϕ, ϕ′ be representations of Λ over k having the same dimension s and let G = (γ ⊗ ϕ)(g), A = (γ ⊗ ϕ)(a), B = (γ ⊗ ϕ)(b), G′ = (γ ⊗ ϕ′)(g), A′ = (γ ⊗ ϕ′)(a), B′ = (γ ⊗ ϕ′)(b). Consider the matrix equalities (in the variable X) GX = XG′, AX = XA′, BX = XB′, (∗) viewing all their matrices as s × s block ones. 18 On modular representations of semigroups Sp × Tp The equalities (of the s × s ij-blocks) (GX)ij = (XG′)ij , (AX)ij = (XA′)ij , (BX)ij = (XB′)ij , i, j ∈ {1, 2, 3, 4} are denoted by (1; ij), (2; ij), (3; ij), respectively. We first write down all equalities of the forms (2; ij) and (3; ij) besides the trivial identities 0 = 0 and Xii = Xii: (2; 1, 1) : X21 = 0, (2; 1, 2) : X22 = X11, (2; 1, 3) : X23 = 0, (2; 1, 4) : X24 = X13, (2; 2, 2) : 0 = X21, (2; 2, 4) : 0 = X23, (2; 3, 1) : X41 = 0, (2; 3, 2) : X42 = X31, (2; 3, 3) : X43 = 0, (2; 3, 4) : X44 = X33, (2; 4, 2) : 0 = X41, (2; 4, 4) : 0 = X43, (3; 1, 2) : X12 = 0, (3; 1, 3) : X13 = 0, (3; 1, 4) : X14 = 0, (3; 2, 1) : 0 = X21, (3; 3, 1) : 0 = X31, (3; 4, 1) : 0 = X41. From these equalities it follows that X =      X11 0 0 0 0 X11 0 0 0 X32 X33 X34 0 0 0 X33      . Then from the equalities (1; 3, 2) : ϕ(y)X11 = X33ϕ′(y), (1; 3, 4) : ϕ(x)X33 = X33ϕ′(x) (∗∗) (the only two nontrivial equalities of the form (1; ij) modulo the equalities (2; ij) and (3; ij)) we have that the matrix k-representations ϕ and ϕ′ of Λ = kΓ are equivalent if so are the matrix k-representations γ ⊗ ϕ and γ ⊗ ϕ′ of S2 × T2 (because X11 and X33 are invertible if so is X). Thus, for the representation γ condition 2) of the definition of wild semigroups holds. From the form of the matrix X it follows that the endomorphism algebra of γ ⊗ ϕ is local if and only if so is the endomorphism algebra of ϕ (these algebras are defined, respectively, by (∗) and (∗∗) with ϕ = ϕ′). Therefore γ⊗ϕ is indecomposable if ϕ is indecomposable, and consequently γ satisfies condition 1) of the mentioned definition too. The theorem is proved. Because as a perfect matrix representation of the quiver Γ over the algebra K ′ 2 = k < x′, y′ > one can take the representation x → ( 0 x′ 1 y′ ) , y → ( 1 0 ) , V. Bondarenko, E. Kostyshyn 19 it follows from the proof of our theorem that the following representation λ of the semigroup S2 × T2 over K ′ 2 is perfect: λ(g) =          1 0 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 x′ 0 0 0 1 1 y′ 0 0 0 0 1 0 0 0 0 0 0 1          , λ(a) =          1 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 1          , λ(b) =          1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0          . References [1] Yu. A. Drozd, Tame and wild matrix problems, Lecture Notes in Math. 832 (1980), pp. 242-258. [2] V. M. Bondarenko, Ju. A. Drozd, Representation type of finite groups, Zap. Nauc̆n. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 71 (1977), pp. 24-42 (in Russian); English trans. in J. Soviet Math. 20 (1982), pp. 2515-2528. [3] L. A. Nazarova, Representations of quivers of infinite type, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), pp. 752-791 (in Russian); English trans. in Math. USSR-Izv. 7 (1973), pp. 749-792. [4] P. Donovan, M.-R. Freislich, The representation theory of finite graphs and associated algebras, Carleton Math. Lecture Notes, No. 5. Carleton University, Ottawa, Ont., 1973, 83 pp. [5] V. M. Bondarenko, E. M. Kostyshyn, Modular representations of the semigroup T2, Nauk. Visn. Uzhgorod. Univ., Ser. Mat. Inform. 22 (2011), pp. 26-34 (in Ukrainian). Contact information V. M. Bondarenko Institute of Mathematics, NAS, Kyiv, Ukraine E-Mail: vitalij.bond@gmail.com E. M. Kostyshyn Department of Mechanics and Mathema- tics, Kyiv National Taras Shevchenko Univ., Volodymyrska str., 64, 01033 Kyiv, Ukraine E-Mail: elina.kostyshyn@mail.ru Received by the editors: 17.07.2013 and in final form 17.07.2013.

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