Indecomposable and irreducible t-monomial matrices over commutative rings

We introduce the notion of the defining sequence of a permutation indecomposable monomial matrix over a commutative ring and obtain necessary conditions for such matrices to be indecomposable or irreducible in terms of this sequence.

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Дата:	2016
Автори:	Bondarenko, V.M., Bortos, M., Dinis, R., Tylyshchak, A.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2016
Назва видання:	Algebra and Discrete Mathematics
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Цитувати:	Indecomposable and irreducible t-monomial matrices over commutative rings / V.M. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 11-20. — Бібліогр.: 4 назв. — англ.

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spelling	irk-123456789-1557432019-06-18T01:25:21Z Indecomposable and irreducible t-monomial matrices over commutative rings Bondarenko, V.M. Bortos, M. Dinis, R. Tylyshchak, A. We introduce the notion of the defining sequence of a permutation indecomposable monomial matrix over a commutative ring and obtain necessary conditions for such matrices to be indecomposable or irreducible in terms of this sequence. 2016 Article Indecomposable and irreducible t-monomial matrices over commutative rings / V.M. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 11-20. — Бібліогр.: 4 назв. — англ. 1726-3255 2010 MSC:15B33, 15A30. http://dspace.nbuv.gov.ua/handle/123456789/155743 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	We introduce the notion of the defining sequence of a permutation indecomposable monomial matrix over a commutative ring and obtain necessary conditions for such matrices to be indecomposable or irreducible in terms of this sequence.
format	Article
author	Bondarenko, V.M. Bortos, M. Dinis, R. Tylyshchak, A.
spellingShingle	Bondarenko, V.M. Bortos, M. Dinis, R. Tylyshchak, A. Indecomposable and irreducible t-monomial matrices over commutative rings Algebra and Discrete Mathematics
author_facet	Bondarenko, V.M. Bortos, M. Dinis, R. Tylyshchak, A.
author_sort	Bondarenko, V.M.
title	Indecomposable and irreducible t-monomial matrices over commutative rings
title_short	Indecomposable and irreducible t-monomial matrices over commutative rings
title_full	Indecomposable and irreducible t-monomial matrices over commutative rings
title_fullStr	Indecomposable and irreducible t-monomial matrices over commutative rings
title_full_unstemmed	Indecomposable and irreducible t-monomial matrices over commutative rings
title_sort	indecomposable and irreducible t-monomial matrices over commutative rings
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/155743
citation_txt	Indecomposable and irreducible t-monomial matrices over commutative rings / V.M. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak // Algebra and Discrete Mathematics. — 2016. — Vol. 22, № 1. — С. 11-20. — Бібліогр.: 4 назв. — англ.
series	Algebra and Discrete Mathematics
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first_indexed	2025-07-14T07:59:05Z
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 22 (2016). Number 1, pp. 11–20 © Journal “Algebra and Discrete Mathematics” Indecomposable and irreducible t-monomial matrices over commutative rings Vitaliy M. Bondarenko, Maria Yu. Bortos, Ruslana F. Dinis, Alexander A. Tylyshchak Communicated by V. V. Kirichenko Abstract. We introduce the notion of the defining sequence of a permutation indecomposable monomial matrix over a commu- tative ring and obtain necessary conditions for such matrices to be indecomposable or irreducible in terms of this sequence. Introduction Let K be a commutative ring (with unity). By a monomial matrix M = (mij) over K we mean a quadratic n × n matrix, in each row and each column of which there is exactly one non-zero element. With such matrix M one can associate the directed graph Γ(M) with n vertices numbered from 1 to n and arrows i → j for all mij 6= 0. Obviously, Γ(M) is the disjoint union of cycles, each of which has the same direction of arrows. If there is only one cycle, the monomial matrix M is called cyclic (in other words, it is a permutation indecomposable monomial matrix). A cyclic matrix of the form M =       0 . . . 0 m1n m21 . . . 0 0 ... . . . ... ... 0 . . . mn,n−1 0       2010 MSC: 15B33, 15A30. Key words and phrases: local ring, similarity, indecomposable matrix, irreducible matrix, canonically t-cyclic matrix, defining sequence, group, representation. 12 Indecomposable and irreducible t-monomial matrices we call canonically cyclic. The sequence v = v(M) = (m21, . . . , mn−1,n, m1n) we call the defining sequence of M , and write M = M(v) = M(m21, . . . , mn−1,n, m1n). The sequence v∗ = v∗(M) = (m1n, mn−1,n, . . . , m21) is called dual to v and the matrix M∗ = M(v∗) dual to M . When all elements mij of a monomial (respectively, cyclic or canonically cyclic) matrix M are of the form tsij (t ∈ K), where sij > 0, the matrix M is called t-monomial (respectively, t-cyclic or canonically t-cyclic); obviously, then tsi 6= 0 for all i. The most interesting cases are, obviously, those when the element t is non-invertible. Matrices of such form were studied by the authors in [1], and in this paper we continue our investigation. 1. Defining sequences and indecomposability Through this section K denotes a commutative local ring with maximal ideal R = Rad K 6= 0 and t ∈ R. All matrices are considered over K. By Es one denotes the identity s × s matrix. 1.1. Permutation similarity. Let M =       0 . . . 0 tsn ts1 . . . 0 0 ... . . . ... ... 0 . . . tsn−1 0       (∗) be a canonically t-cyclic matrix. A permutation of the members xi = tsi of v(M) of the form xi, xi+1, . . . , xm, x1, x2, . . . , xi−1, is called a cyclic permutation. Two matrices M(v) and M(v′) is called cyclically similar if v can be obtained from v′ by a such permutation. It is easy to prove the following statement1. Proposition 1. a) Two canonically t-cyclic matrices is permutation similar if and only if they are cyclically similar. b) The matrix transpose to a canonically t-cyclic matrix is permutation similar to the dual one. 1This statement is valid for matrices with elements from any set if one modifies the definitions accordingly. V. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak 13 1.2. Conjecture. Let M(v) be a canonically t-cyclic n×n matrix with defining sequence v = (x1, x2, . . . , xn), where xi = tsi (see (∗)). The sequence v is called periodic with a period 0 < p < n if p\|n and xs+p = xs for any 1 6 s 6 n−p, and non-periodic if otherwise. In the case of v to be periodic, the matrix M(v) can be reduced by a permutation of its rows and column to the following block-monomial form: N = (Nij) m i,j=1, m = n/p, where N21 = x2Em, N32 = x3Em, . . . , Nn,n−1 = xnEm and N1m = x1M(1, 1, . . . , 1) with 1 to occur m times2. The m × m matrix M(1, 1, . . . , 1) can be indecomposable or decomposable depending on properties of the ring K, and therefore so can be the matrix M(v). Conjecture 6 (V. M. Bondarenko, Private Communication). Any canon- ically t-cyclic matrix over K with non-periodic defining sequence is inde- composable. It is obvious that the idea of a proof of this conjecture basing on decomposition of a given matrix M into a direct sum of others ones, with a final contradiction, is futile. It is most likely the best idea is to use the simple fact that M is indecomposable if any idempotent matrix X such that MX = XM is identity or zero. The main difficulty in this way is that the non-periodicity of the defining sequence of M is not determined by its local properties. In the next subsection we consider a special case in which the condition of the non-periodicity is satisfied automatically. 1.3. 2-homogeneous defining sequences. Let M(v) be again a canonically t-cyclic n×n matrix with defining sequence v =(x1, x2, . . . , xn), where xi = tsi . The sequence v is said to be 2-homogeneous if it is trans- lated by a cyclic permutation on that of the form (a, a, . . . , a, b, b, . . . , b), where a and b 6= a both actually occur. The following theorem proves Conjecture 1 for this case. Theorem 1. Any canonically t-cyclic matrix over K with 2-homogeneous defining sequence is indecomposable. Proof. Let M = M(v) be a canonically t-cyclic n×n matrix and let v has s coordinates equal to a = tp and the other ones equal to b = tq. Assume without loss of generality that v = (a, a, . . . , a, b, b, . . . , b) and p < q. So v = tpv0, where v0 = (1, 1, . . . , 1, tq−p, tq−p, . . . , tq−p). After replacing v 2To do it, one is to arrange the rows and columns in the order 2, 2+p, 2+2p,. . . , 2+(m- 1)p, 3, 3+p, 3+2p,. . . , 3+(m-1)p,. . . , p, 2p, 3p,. . . , mp, p+1, (p+1)+p, (p+1)+2p,. . . , (p+1)+(m-2)p,1. 14 Indecomposable and irreducible t-monomial matrices by v0 and tq−p by t (again without loss of generality), we come to the situation where M =              0 . . . 0 0 . . . 0 t 1 . . . 0 0 . . . 0 0 ... . . . ... ... . . . ... ... 0 . . . 1 0 . . . 0 0 0 . . . 0 t . . . 0 0 ... . . . ... ... . . . ... ... 0 . . . 0 0 . . . t 0              with s elements to equal 1 and k = n − s elements equal to t. Arrange the rows and columns of the matrix M in the order 1, 2, . . . , n − k, n, n − 1, . . . , n − k + 2, n − k + 1, denoting the new matrix by N : N = ( N11 N12 N21 N22 ) =                 0 . . . 0 0 t 0 . . . 0 1 . . . 0 0 0 0 . . . 0 ... . . . ... ... ... ... . . . ... 0 . . . 1 0 0 0 . . . 0 0 . . . 0 0 0 t . . . 0 ... . . . ... ... ... ... . . . ... 0 . . . 0 0 0 0 . . . t 0 . . . 0 1 0 0 . . . 0                 . Prove that there is not a non-trivial idempotent matrix commuting with N (see the previous subsection). Let C be an n × n matrix such that NC = CN , where C = ( C11 C12 C21 C22 ) = =            c11 . . . c1,n−k c1,n−k+1 . . . c1n ... . . . ... ... . . . ... cn−k,1 . . . cn−k,n−k cn−k,n−k+1 . . . cn−k,n cn−k+1,1 . . . cn−k+1,n−k cn−k+1,n−k+1 . . . cn−k+,n ... . . . ... ... . . . ... cn1 . . . cn,n−k cn,n−k+1 . . . cnn            with the same partition as N . V. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak 15 We denote by (i, j) the scalar equality (NC)ij = (CN)ij . All compa- risons below are considered modulo the maximal ideal R = Rad K (that is, they are equalities over the residue field K/R). We use by default the following simple fact: tx = ty implies x ≡ y (x, y ∈ K). Since every jth column of CN with j > n − k consists of elements from tK, we have from the equations (i, j) for i = 2, 3, . . . , n − k and i = n that C ≡ ( C11 0 C21 C22 ) = =            c11 . . . c1,n−k 0 . . . 0 ... . . . ... ... . . . ... cn−k,1 . . . cn−k,n−k 0 . . . 0 cn−k+1,1 . . . cn−k+1,n−k cn−k+1,n−k+1 . . . cn−k+1,n ... . . . ... ... . . . ... cn1 . . . cn,n−k cn,n−k+1 . . . cnn            . The equations (i, j) for i, j = 1, 2, . . . , n − k mean that the matrix C11 commutes modulo R with the lower Jordan block N11 and hence the matrix C11 is lower triangular modulo R (see, e. g., [2, Chap. VIII] or [3, Theorem 3.2.4.2]). Further, from the equlities (n − k + i, n − k + j + 1) : tCn−k+i+1,n−k+j+1 = tCn−k+i,n−k+j for 1 6 i < j 6 k − 1 it follows that all elements of the matrix C22 belonging to its lth upper diagonal3 are pairwise comparable modulo R, 1 6 l 6 k − 2. But since the equalities (n − k + i, n − k) : tCn−k+i+1,n−k = Cn−k+i,n, 1 6 i 6 k − 1, imply that the last elements of all (mentioned above) upper diagonals are comparable with 0, we have eventually that the matrix C22 is, as well as C11 (see above), upper triangular modulo R. Finally, from the equalities I. (2, 1) : c11 = c22, (3, 2) : c22 = c33, . . ., (n − k, n − k − 1) : cn−k−1,n−k−1 = cn−k,n−k, II. (n − k + 1, n − k + 2) : tcn−k+1,n−k+1 = tcn−k+2,n−k+2, (n − k, n − k + 1) : tcn−k,n−k = tcn−k+1,n−k+1, . . . , (n − 1, n) : tcn−1,n−1 = tcn,n, 3The lth upper diagonal of a matrix M = (mij), where l > 1, is the collection of elements mi,i+l. 16 Indecomposable and irreducible t-monomial matrices III. (n, n − k) : cn−k,n−k = cn,n, it follows that c11 ≡ c22 ≡ · · · ≡ cnn. Thus we prove that the matrix C is comparable to an upper triangular one with the same elements on the main diagonal. It easily follows that if C2 ≡ C, then C ≡ En or C ≡ 0, and, consequently, because the comparisons are modulo the only maximal ideal of K, C = En or C = 0, respectively. 1.4. Applications in the representation theory of groups. Through this subsection K is as above and of characteristic ps (p is simple, s > 1). All groups G are assumed to be finite of order \|G\| > 1. The number of nonequivalent indecomposable matrix K-representations of degree n of a group G is denoted by indK(G, n). From [4] it follows that indK(G, n) > \|K/R\| for any p-group G of order \|G\| > 2 and n > 1. Here we strengthen this result in the case of both cyclic groups and radicals. Theorem 2. Let R = tK 6= 0 with t being nilpotent. Then, for a cyclic p-group G of some order N (hence of greater order), indK(G, n) > n − 1 for any n > 1. Proof. Let S be an n × n matrix over K that is nilpotent modulo R; then Sn ≡ 0 (mod R) and S2n ≡ 0 (mod R2). It is easy to see that the map ΓS : a → ΓS(a) = En + uS with u = tm−2 is a K-representation of a cyclic group G = 〈a〉 of an order pr > 2n, r > 2. It is indecomposable if and only if so is modulo Ann u := {x ∈ K \| tm−2x = 0} = R2, and representations ΓS and ΓS′ are equivalent if and only if the matrices S and S′ are similar modulo R2. Consider the K-representations ΓMk , k = 1, 2, . . . , n − 1, with the matrices Mk = M(1, . . . , 1, t . . . , t), where 1 occurs k times (see the previous subsection). They are non-equivalent, because Mk has rank k modulo R, and indecomposable by Theorem 1 (taking into account that Rad (K/Ann u) = R/R2 is a principal ideal of K/R2 generated by t + R2 6= R2). Theorem 3. Let the characteristic of K is p and R = tK 6= 0 with t2 = 0. Then, for any cyclic p-group G and n > \|G\|, indK(G, n) > \|G\| − 2. Proof. We use the notation of the proof of Theorem 2. It is easy to see that, for 0 < k < n, the map Λk : a → Λ(a) = En + Mk is a K- representation of a cyclic group G = 〈a〉 if k + 2 6 \|G\|; in particular, if 0 < k < \|G\| − 1 6 n − 1. By the last condition, their number is equal to V. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak 17 \|G\| − 2. The representations Λk are indecomposable by Theorem 1, and non-equivalent (see the previous proof). 2. Defining sequences and irreducibility Through this section K also denotes a commutative local ring with maximal ideal R = Rad K 6= 0; t 6= 0 is any element of R unless otherwise stated. For an n × n matrix A over K, we denote by [i a → j]+ (resp. [i a → j]−) the following similarity transformation of A: adding ith row (resp. column), multiplied by a, to jth row (resp. column), and then subtracting jth column (resp. rows), multiplied by a, from ith column (resp. row). 2.1. Theorems on irreducible canonically t-cyclic matrices. Let M = M(v) be a canonically t-cyclic n × n matrix with defining sequence v = (x1, x2, . . . , xn), where xi = tsi (see (∗)). We call the sequence v0 = v0(M) = (s1, s2, . . . , xn) the weighted sequence of M , and the number w = w(M) = s1 + s2 + · · · + sn the weight of M4. From the results of [1] it follows the next theorem5. Theorem 4. If the matrix M(v) is irreducible, then its weight is prime to n. Corollary 1. If t2 = 0 and the matrix M(v) is irreducible, then its rank modulo R is prime to n. By a connected subsequence of the length 1 6 l 6 n of a defining sequence v = (x1, x2, . . . , xn) we mean any subsequence which maps by a cyclic permutation on one of the form x1, x2, . . . , xl. If it is 2-homogeneous i. e., by analogy with the above said, has the form u = (ti, ti, . . . , ti, tj , tj , . . . , tj), ti 6= tj , where ti and tj both actually occur, then the pair (p, q), consisting of the numbers of occurrences of ti and tj we call the type of u. We shall obtain the following theorem as a consequence of statements in more general situations. Theorem 5. If the matrix M(v) with 2-homogeneous defining sequence v is irreducible and n > 5, t2 = 0, then its weight is equal 1, 2 or n − 1. Concerning the cases when the weight is equal 1, n − 1 see [1, Intro- duction]. 4One can write M(t, v0) instead of M(v) (as in [1]). 5A quadratic matrix is irreducible if it is not similar to a 2×2 upper block triangular matrix with quadratic diagonal blocks. 18 Indecomposable and irreducible t-monomial matrices 2.2. Defining sequences with subsequences of type (2, 4) and (4, 2). We are interested in the cases when a 2-homogeneous subse- quence has the form (ts, ts, ts, ts, 1, 1) or (1, 1, ts, ts, ts, ts). Since they are mutually dual (see Proposition 1), we consider only the first case. Proposition 2. If tm = 0 and a canonically t-cyclic n × n matrix M(v) is irreducible, then the sequence v does not contain a subsequence of the form (ts, ts, ts, ts, 1, 1) with m 6 2s < 2m. We prove a more general statement replacing (ts, ts, ts, ts, 1, 1) by (ti, tj , tp, tq, 1, 1) with 0 < i, j, p, q < m, i + j > m, p + q > m, assuming (by Proposition 1) that the subsequence is the beginning of v and (by Theorem 4) that n > 6. This follows from the following: if we perform with the reducible matrix N =                  0 0 . . . 0 0 −tj 1 0 0 1 0 . . . 0 0 0 0 0 0 0 α1 . . . 0 0 0 0 0 0 ... ... . . . ... ... ... ... ... ... 0 0 . . . αk 0 0 0 0 0 0 0 . . . 0 ti 0 0 0 0 0 0 . . . 0 0 0 0 0 tq 0 0 . . . 0 0 0 1 0 0 0 0 . . . 0 0 0 0 tp 0                  (k = n − 6) the transformation [(n − 2) tj → (n − 3)]−, [1 −1 → (n − 1)]+, [2 −tp → n]+, and arrange the rows and columns of the resulting monomial matrix N ′ =                  0 0 . . . 0 0 0 1 0 0 1 0 . . . 0 0 0 0 0 0 0 α1 . . . 0 0 0 0 0 0 ... ... . . . ... ... ... ... ... ... 0 0 . . . αk 0 0 0 0 0 0 0 . . . 0 ti 0 0 0 0 0 0 . . . 0 0 0 0 0 tq 0 0 . . . 0 0 tj 0 0 0 0 0 . . . 0 0 0 0 tp 0                  in the order n − 4, n − 3, n − 1, n, n − 2, 1, 2, . . . , n − 5, we get the matrix M(ti, tj , tp, tq, 1, 1, α1, . . . , αk). V. Bondarenko, M. Bortos, R. Dinis, A. Tylyshchak 19 2.3. Defining sequences with subsequences of type (3, 3). We consider the cases, when a 2-homogeneous subsequence has the form (ts, ts, ts, 1, 1, 1) or (1, 1, 1, ts, ts, ts), in the same way as those in subsection 2.2; therefore we present only the main part and do not repeat similar assumptions and comments. Proposition 3. If tm = 0 and a canonically t-cyclic n × n matrix M(v) is irreducible, then the sequence v does not contain a subsequence of the form (ts, ts, ts, 1, 1, 1) with m 6 2s < 2m. We prove a more general statement replacing (ts, ts, ts, 1, 1, 1) by (ti, tj , tp, 1, 1, 1) with 0 < p, q, j < m, i + j > m, 2p > m. This follows from the following: if we perform with the reducible matrix N =                  0 0 0 . . . 0 0 −tj 1 0 1 0 −tp . . . 0 0 0 0 0 0 1 0 . . . 0 0 0 0 0 0 0 α1 . . . 0 0 0 0 0 ... ... ... . . . ... ... ... ... ... 0 0 0 . . . αk 0 0 0 0 0 0 0 . . . 0 ti 0 0 0 0 0 0 . . . 0 0 0 0 tp 0 0 0 . . . 0 0 0 1 0                  (k = n − 6) the transformation [(n − 1) tj → (n − 2)]−, [1 −1 → n]+, [2 −tp → (n − 1)]+, [3 −tp → 1]+, and arrange the rows and columns of the resulting monomial matrix N ′ =                  0 0 0 . . . 0 0 0 1 0 1 0 0 . . . 0 0 0 0 0 0 1 0 . . . 0 0 0 0 0 0 0 α1 . . . 0 0 0 0 0 ... ... ... . . . ... ... ... ... ... 0 0 0 . . . αk 0 0 0 0 0 0 0 . . . 0 ti 0 0 0 0 0 0 . . . 0 0 0 0 tp 0 0 0 . . . 0 0 tj 0 0                  in the order n − 3, n − 2, n, n − 1, 1, 2, . . . , n − 4, we get the matrix M(ti, tj , tp, 1, 1, 1, α1, α2, . . . , αk). 20 Indecomposable and irreducible t-monomial matrices 2.4. Proof of Theorem 5. The proof follows from Propositions 2 and 3. Indeed, the first proposition implies at once that M = M(v) is reducible, if w(M) = n−2, and the second one that M = M(v) is reducible, if 2 < w(M) < n − 2 (in both the cases it need to take m = 2, s = 1). In conclusion, we note that in the cases n = 2, 3 the theorem is trivial, in the case n = 4 it follows from Theorem 4 and in the case n = 5 there is the only exception, namely the matrix M(1, 1, t, t, t, ) of weight 3 is irreducible. The theorem is proved. References [1] V. M Bondarenko, M. Yu. Bortos, M. Yu. Dinis, A. A. Tylyshchak, Reducibility and irreducibility of monomial matrices over commutative rings, Algebra Discrete Math. 16 (2013), no. 2, pp. 171–187. [2] F. R. Gantmaher, The theory of matrices, FIZMATLIT, 2004, 560 p. [3] R. A. Horn C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013, 643 p. [4] P. M. Gudivok, I. B. Chukhraj, On the number of indecomposable matrix rep- resentations of given degree of a finite p-group over commutative local rings of characteristic ps, Nauk. Visn. Uzhgorod Univ. Ser. Mat. Inform. 5 (2000), pp. 33–40 (in Ukrainian). Contact information V. M. Bondarenko, M. Yu. Bortos Institute of Mathematics, Tereshchenkivska str., 3, 01601 Kyiv, Ukraine E-Mail(s): vit-bond@imath.kiev.ua, bortosmaria@gmail.com Web-page(s): http://www.imath.kiev.ua R. F. Dinis Faculty of Mechanics and Mathematics, Kyiv Na- tional Taras Shevchenko Univ., Volodymyrska str., 64, 01033 Kyiv, Ukraine E-Mail(s): ruslanadinis@ukr.net A. A. Tylyshchak Faculty of Mathematics, Uzhgorod National Univ., Universytetsyka str., 14, 88000 Uzhgorod, Ukraine E-Mail(s): alxtlk@gmail.com Received by the editors: 29.08.2016.

Indecomposable and irreducible t-monomial matrices over commutative rings

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