Orthoscalar representations of the partially ordered set (N, 4)

We obtain a one-parameter series of orthoscalar representations of the partially ordered set (N, 4). This proves that the classification of such representations is a problem of infinite type.

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Datum:	2012
Hauptverfasser:	Kruglyak, S.A., Livinsky, I.V.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2012
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spelling	irk-123456789-1522392019-06-10T01:25:36Z Orthoscalar representations of the partially ordered set (N, 4) Kruglyak, S.A. Livinsky, I.V. We obtain a one-parameter series of orthoscalar representations of the partially ordered set (N, 4). This proves that the classification of such representations is a problem of infinite type. 2012 Article Orthoscalar representations of the partially ordered set (N, 4) / S.A. Kruglyak, I.V. Livinsky // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 217–229. — Бібліогр.: 18 назв. — англ. 1726-3255 2010 MSC:16G20. http://dspace.nbuv.gov.ua/handle/123456789/152239 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 obtain a one-parameter series of orthoscalar representations of the partially ordered set (N, 4). This proves that the classification of such representations is a problem of infinite type.
format	Article
author	Kruglyak, S.A. Livinsky, I.V.
spellingShingle	Kruglyak, S.A. Livinsky, I.V. Orthoscalar representations of the partially ordered set (N, 4) Algebra and Discrete Mathematics
author_facet	Kruglyak, S.A. Livinsky, I.V.
author_sort	Kruglyak, S.A.
title	Orthoscalar representations of the partially ordered set (N, 4)
title_short	Orthoscalar representations of the partially ordered set (N, 4)
title_full	Orthoscalar representations of the partially ordered set (N, 4)
title_fullStr	Orthoscalar representations of the partially ordered set (N, 4)
title_full_unstemmed	Orthoscalar representations of the partially ordered set (N, 4)
title_sort	orthoscalar representations of the partially ordered set (n, 4)
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2012
url	http://dspace.nbuv.gov.ua/handle/123456789/152239
citation_txt	Orthoscalar representations of the partially ordered set (N, 4) / S.A. Kruglyak, I.V. Livinsky // Algebra and Discrete Mathematics. — 2012. — Vol. 14, № 2. — С. 217–229. — Бібліогр.: 18 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT kruglyaksa orthoscalarrepresentationsofthepartiallyorderedsetn4 AT livinskyiv orthoscalarrepresentationsofthepartiallyorderedsetn4
first_indexed	2025-07-13T02:37:39Z
last_indexed	2025-07-13T02:37:39Z
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fulltext	Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 14 (2012). Number 2. pp. 217 – 229 c© Journal “Algebra and Discrete Mathematics” Orthoscalar representations of the partially ordered set (N, 4) S. A. Kruglyak, I. V. Livinsky Communicated by Yu. A. Drozd Abstract. We obtain a one-parameter series of orthoscalar representations of the partially ordered set (N, 4). This proves that the classification of such representations is a problem of infinite type. 1. Introduction Many problems of functional analysis can be formulated and solved in terms of the theory of representations of ∗-quivers and ∗-algebras. Representations, in Hilbert spaces, of ∗-algebras with self-adjoint gen- erators whose sum is a multiple of the identity and whose spectra are fixed were studied in numerous works (see, e.g., [1–3]). They are naturally associated with orthoscalar representations of certain ∗-quivers (or graphs) investigated in [4–7]. Collections of operators with special fixed spectra and the sum equal to the identity operator that are associated with the extended Dynkin graphs D̃4, Ẽ6, and Ẽ7 were studied in [1, 3, 8, 9]. Some results on representations of algebras associated with Ẽ8 are presented in [10]. In [11–12] infinite two-parameter series of irreducible representations of graphs Ẽ6, Ẽ7, and Ẽ8 with above-mentioned special characters were constructed explicitly (canonical forms of such representations were presented). Representations of partially ordered sets (posets) were introduced by L. A. Nazarova and A. V. Roiter in [13], where and algorithm was 2010 MSC: 16G20. Key words and phrases: partially ordered set, orthoscalar representation, infinite type. Jo ur na l A lg eb ra D is cr et e M at h. 218 Orthoscalar representations of the set (N, 4) constructed allowing to find out whether a certain poset has finitely or infinitely many indecomposable representations. Kleiner, in his paper [14], proved using this algorithm that a poset is of finite type if and only if it does not contain “critical” subsets: (1, 1, 1, 1), (2, 2, 2), (1, 3, 3), (1, 2, 5) and (N, 4) = {a1 < a2 > b1 < b2; c1 < c2 < c3 < c4} (here (l1, l2, ..., lm) denotes the cardinal sum of chains of lengths l1, l2, ..., lm). Results on finite representation type of quivers were translated to finite dimensional Hilbert (unitary) spaces in [4]; the analogue of Gabriel’s theorem was proved for quivers and their orthoscalar representations. For the proof of Kleiner’s theorem analogue for orthoscalar represen- tations of posets it should be proved, in particular, that the classification of critical posets is a problem of infinite type. For primitive posets this problem reduces (see [15]) to the similar problem regarding extended Dynkin graphs. In the present paper it is proved that the classification of orthoscalar representations of the last critical poset (N, 4) is a problem of infinite type. For another definition of orthoscalar representations of posets and Kleiner’s theorem in this treatment, see [17–18]. 2. Notation and auxiliary facts Recall some notation and facts related to orthoscalar representations of quivers [4–6]. A quiver Q with a set of vertices Qv, \|Qv\| = N and a set of arrows Qa is called divided if Qv = ◦ Q ⊔ • Q and, for any α ∈ Qa, its origin tα belongs to ◦ Q and the end hα belongs to • Q. One says that the quiver Q is of multiplicity one if, for α 6= β, one has either tα 6= tβ or hα 6= hβ . The vertices from ◦ Q and • Q are called even and odd respectively. Let m = \| • Q\|, n = \| ◦ Q\|, • Q = {i1, i2, ..., im}, ◦ Q = {j1, j2, ..., jn}. A representation T of a quiver Q associates a vertex i ∈ Qv with a vector space T (i) and an arrow α : j → i, α ∈ Qa, with a linear mapping Tij : T (j)→ T (i). A representation T of a divided quiver of multiplicity one with fixed bases of spaces T (i), i ∈ Qv can be associated with a matrix divided into m horizontal and n vertical strips, i.e., with a matrix T = [Til,jk ]k=1,n l=1,m . We assume that Til,jk = 0 if there does not exist α ∈ Qa such that tα = jk, hα = il. Let −→ Ti = [Ti,j1 \|Ti,j2 \| ... \|Ti,jn ], Jo ur na l A lg eb ra D is cr et e M at h. S. A. Kruglyak, I. V. Livinsky 219 T ↓ j =   Ti1,j ... Tim,j   , −→ Ti : n⊕ k=1 T (jk)→ T (i), T ↓ j : T (j)→ m⊕ l=1 T (il), A divided quiver of multiplicity one is called ordered, if • Q and ◦ Q are posets. A representation T of an ordered divided quiver Q of multiplicity one is called orthoscalar 1 if the spaces T (i), i ∈ Qv are finite dimensional Hilbert (unitary) spaces (over the field of complex numbers C), every i ∈ Qv is associated with a positive real number χi, and the following conditions are satisfied: 1) −→ Ti · −→ Ti ∗ = χiIi for i ∈ • Q; T ↓∗ j · T ↓ j = χjIj for j ∈ ◦ Q; 2) if i′ < i′′, i′, i′′ ∈ • Q, then χi′ > χi′′ and −→ Ti′ · −→Ti′′ ∗ = 0; if j′ < j′′, j′, j′′ ∈ ◦ Q, then χj′ > χj′′ and T ↓∗ j′ · T ↓ j′′ = 0. If m = 1, a representation T of an ordered quiver Q is called an orthoscalar representation of a poset. With an orthoscalar representation T of an ordered divided quiver of multiplicity one we associate two N -dimensional vectors (N = m + n): the dimension d of the representation T , d = {d(j)}j∈Qv , where d(j) = dim T (j), and the character χ of the representation T, χ = {χ(j)}j∈Qv , χ(j) = χj is defined above. It is easy to see that m∑ l=1 d(il)χ(il) = n∑ k=1 d(jk)χ(jk). (1) Indeed, the space of rows −→x = (x1, x2, ..., xs) over the field of complex numbers with the dot product (−→x ,−→y ) = (x1y1 + ... + xsys) is a unitary space. The norm \|\|−→x \|\| of a row −→x is defined as √ (−→x ,−→x ), two rows are orthogonal if (−→x ,−→y ) = 0. The unitary space of columns (and the column norm \|\|y↓\|\|) are defined similarly. Then, equality (1) for the matrix of representation T = [zij ] means that the sum of squares of row norms for the matrix T is equal to the sum of squares of column norms for the matrix T , and is equal to ∑ ij zijzij . 1Definition belongs to A. V. Roiter. Jo ur na l A lg eb ra D is cr et e M at h. 220 Orthoscalar representations of the set (N, 4) The conditions of orthoscalarity, in particular, mean that the rows in a single (ith) horizontal strip of a block matrix T are orthogonal and have constant norm ( √ χi), and if i < j in • Q then the rows of the ith and jth horizontal strips are orthogonal. Similar properties hold for the columns of T . A non-ordered divided quiver of multiplicity one can be considered as a special case of an ordered quiver (in ◦ Q and • Q all elements assigned to be incomparable). Let Rep Q be the category of representations of a non-ordered quiver Q whose objects are representations T and a morphism of a representation T to a representation T̃ is defined as a family of linear mappings C = {Ci}i∈Qv , Ci : T (i) → T̃ (i), such that, for every α ∈ Qv with tα = j, hα = i, the diagram T (j) T (i) T̃ (j) T̃ (j) ✲T (α) ❄ Cj ❄ Ci ✲T̃ (α) (2) is commutative, i.e., CiTij = T̃ijCj . Define the matrices A = diag {Ci1 , ..., Cim}, B = diag {Cj1 , ..., Cjn}. Then the commutativity of the diagram (2) implies AT = T̃B. (3) In what follows, we also use the notation C = (A, B). Two representa- tions T and T̃ are equivalent if there exists an invertible morphism from T to T̃ (with the matrices A and B being invertible). Define the category Repos Q of orthoscalar representations of a non- ordered divided quiver Q of multiplicity one as a subcategory of Rep Q, whose objects are orthoscalar representations of Q and whose morphisms are morphisms C = {Ci}i∈Qv from Rep Q, such that in addition to the commutativity of diagrams (2), the diagram T (j) T (i) T̃ (j) T̃ (j) ❄ Cj ✛ T (α)∗ ❄ Ci ✛ T̃ (α)∗ (4) Jo ur na l A lg eb ra D is cr et e M at h. S. A. Kruglyak, I. V. Livinsky 221 is also commutative, i.e., AT = T̃B and BT ∗ = T̃ ∗A. (5) Let S be a poset, i.e., for some ordered quiver Q we have m = 1, • Q = S. With a representation T we associate a matrix T = [Tj1 \|Tj2 \| ... \|Tjn ], Tjk ≡ Ti1, jk : T (jk)→ T (i1). (6) The orthoscalarity of the representation T of the partially ordered set S means that a) T ∗ jk Tjk = χjk Ijk , k = 1, ..., n, (7) b) T ∗ jk Tjl = 0 for jk < jl, and χjk > χjl , (8) c) n∑ k=1 Tjk T ∗ jk = χi1 I. (9) The representation T could be considered also as an orthoscalar representation of the quiver Q j2 . . . jn−1 j1 i1 jn ❅ ❅ ❅❘ � � �✠ ✲ ✛ Define the category Repos S as a full subcategory of Repos Q, whose objects are orthoscalar representations of a partially ordered set S (i.e., a morphism C ′ : T → T̃ in the category Repos S is defined as a pair of matrices (A, B), where A = Ci, B = diag {Cj1 , ..., Cjn}, such that equalities (5) hold). It was proved (see, e.g., [16]) that T and T̃ are equivalent in Repos Q, Repos S if and only if they are unitarily equivalent, i.e., an invertible morphism C consists of unitary matrices Ci, Cj . Decomposable repre- sentations are defined in a natural way; if T = T1 ⊕ T2 in the category Repos Q then T1(i)⊕ T2(i) is the orthogonal sum of unitary spaces. A representation T is called a Schur (brick) representation in the cate- gory Repos Q if its endomorphism ring in this category is one-dimensional (isomorphic to C). As is known, a representation T is indecomposable in the category Repos Q if and only if it is a Schur representation (see, e.g., [6], Note 4). Jo ur na l A lg eb ra D is cr et e M at h. 222 Orthoscalar representations of the set (N, 4) 3. Orthoscalar representations of the partially ordered set (N, 4) Hence, let S be a partially ordered set with a Hasse diagram a2 b2 a1 b1 c4 c3 c2 c1 (10) i.e., the set (N, 4). Let Q be a quiver corresponding to S, ◦ Q = {a1, a2, b1, b2, c1, c2, c3, c4} = S, • Q = {i1}: a2 . . . c3 a1 i1 c4 ◗ ◗◗s ✑ ✑✑✰ ✲ ✛ Let T be an indecomposable orthoscalar representation of a poset S in the dimension d = {di1 ; da1 , ..., dc4 }. Assume that the dimension is the following: 1 2 2 1 1 1 1 1 , di1 = dim T (i1) = 5. (11) (we arrange the dimensions of representation spaces in accordance with the location of the vertices of the Hasse diagram for visibility). Fix the character χ = {χi1 ; χa1 , . . . , χc4 } of the representation T : Jo ur na l A lg eb ra D is cr et e M at h. S. A. Kruglyak, I. V. Livinsky 223 2 2 3 3 1 2 3 4 , χi1 = 5. (12) Prior to the calculation of matrix elements of the representation T , with the use of relations (7) – (9) we reduce the representation to the “canonical” form by using admissible unitary transformations (T̃ = UiTijV ∗ j ). We reduce some matrix elements to zero elements, and some nonzero elements to positive or negative elements (by the multiplication of a row or a column of the matrix by a certain number eiϕ; this is a unitary matrix transformation2). In this reduction, for simplicity, we use the following notation: The symbol 0\|k at any place of the matrix T means that one can obtain a zero element at this place in the kth step with the use of unitary transformations of the rows of the horizontal strip and the columns of the vertical strip that correspond to this place. The symbol 0k means that a zero element is obtained solely with the use of unitary transformations of columns. The symbol 0\|k means that a zero element is obtained solely with the use of unitary transformations of rows. The symbol −→ 0k means that a zero element is obtained due to the orthogonality of columns of the vertical strip (or of two distinct strips, comparable in the sense of partial order), and the symbol 0↓k means that a zero element is obtained due to the orthogonality of rows of the horizonal strip (or of two distinct strips, comparable in the sense of partial order). Moreover, while obtaining a zero element on the kth step, we do not “spoil” the zero elements obtained earlier. The symbol a+ ij (a− ij) means that an element at the indicated place is made positive (negative). We hope that the step-by-step reduction process can be easily reproduced. Furthermore, embed our representation to another matrix problem for which it is easier to obtain the “canonical” form and calculate matrix elements. 2The reduction technique of representation matrices for an orthoscalar representation construction of a fixed dimension belongs to L. A. Nazarova. Jo ur na l A lg eb ra D is cr et e M at h. 224 Orthoscalar representations of the set (N, 4) Consider the orthoscalar representation Γ of the quiver P (two numbers located at every vertex are the dimension of the representation space corresponding to it and the corresponding character value in parentheses): s4 1(3) �� t1 1(1) s1 2(2) oo // t2 3(3) s2 4(4) oo // t3 5(5) s5 3(3) oo // t4 1(1) s3 2(2) OO ◦ P = {s1, s2, s3, s4, s5}, • P = {t1, t2, t3, t4}. Denote by Asi,ti ≡ Aij the matrix blocks of the representation Γ. Then Γ =   A11 A21 A22 A32 A33 A34 A35 A45   (empty cells contain zero elements). Denote the matrix elements of Γ by aij ; therefore, Γ = [aij ] is a matrix of a “general” dimension 10× 12. Reduce the representation T of the poset S, T = [Tc1 \|Tc2 \|Tc3 \|Tc4 \|Tb2 \|Tb1 \|Ta2 \|Ta1 ] to T =   a53 a54 a55 a56 ←− 02 ←− 02 a+ 59 −→ 02 02 a5,12 a63 a64 a65 a66 a67 a68 0\|1 a6,10 a6,11 a6,12 a73 a74 a75 a76 a77 a78 0\|1 a7,10 a7,11 a7,12 a83 a84 a85 a86 a87 a88 0\|1 a8,10 a8,11 a8,12 a93 a94 a95 a96 a97 a98 0\|1 a9,10 a9,11 a9,12   and embed it into Γ in the following way (the indices at the matrix elements of T correspond to their future location in the matrix Γ): A32 = [Tc1 Tc2 Tc3 Tc4 ], (Tci are united in a single block), A33 = Tb2 , A34 = Tb1 , A35 = [Ta2 Ta1 ] (Tai are united in a single block) A45 = [a10,10 a10,11 0]. If the matrices A11, A21, A22, A45 are diagonalized with the use of admissible unitary transformations, then the reduction problem of remain- ing matrices A32, A33, A34, A35 coincides with the reduction problem Jo ur na l A lg eb ra D is cr et e M at h. S. A. Kruglyak, I. V. Livinsky 225 of the representation T of the poset S. Moreover, two representations Γ and Γ̃ are unitarily equivalent if and only if the embedded into them representations T and T̃ of the poset S are unitarily equivalent. The result of the reduction (and the reduction process described in our notation) is the following: [ A32 A33 A34 A35 0 0 0 A45 ] = =   a+ 53 03 03 03 ←− 02 ←− 02 a+ 59 −→ 02 03 a+ 5,12 a− 63 a+ 64 06 06 a+ 67 06 0\|1 ←− 05 ←− 05 a+ 6,12 0↓5 a+ 74 a+ 75 09 a− 77 −→ 09 0\|1 a+ 7,10 010 0\|4 0↓5 0↓7 a+ 85 a+ 86 0↓8 a+ 88 0\|1 a− 8,10 a+ 8,11 0\|4 0↓5 0↓7 0\|13 ac 96 0↓8 ac 98 0\|1 0↓13 a+ 9,11 0\|4 0 0 0 0 0 0 0 a+ 10,10 a+ 10,11 0   , here ac ij means that the element aij is a complex number; [ A11 0 A21 A22 ] =   a+ 11 a+ 12 0 0 0 0 a+ 21 016 a+ 23 a+ 24 −→ 012 −→ 012 a− 31 a+ 32 0\|11 a+ 34 a− 35 −→ 015 0\|17 a+ 42 0\|11 0\|14 a+ 45 a− 46   . The sense of embedding the representation T to the representation Γ is in the idea that the matrix Γ in the “canonical” form is more “sparse” in the number and location of zeros; this allows to find the matrix elements by a fixed character of the representation. We show that the present representation depends on two real pa- rameters (t and p), i.e., all the matrix elements can be expressed with these parameters, using the condition of orthoscalarity only. At every step we will need to solve either a linear equation (obtained from the row of column orthogonality) or a quadratic equation (obtained from the equality of a row or column norm to a fixed character value). We present the construction order of the matrix elements showing the condition near every non-zero element that allows to find it. Jo ur na l A lg eb ra D is cr et e M at h. 226 Orthoscalar representations of the set (N, 4) The construction order of the elements aij 1) a+ 59 — calculating the norm of the 9th column; 2) a+ 53 = √ t, t — a parameter; 3) a+ 5,12 — calculating the norm of the 5th row; 4) a+ 6,12 — calculating the norm of the 12th column; 5) a− 63 — from the orthogonality condition of the 5th and the 6th rows; 6) a+ 23 — calculating the norm of the 3rd column; 7) a+ 67 = √ p, p — another parameter; 8) a+ 64 — calculating the norm of the 6th row; 9) a+ 24 — from the orthogonality condition of the 3rd and the 4th columns; 10) a+ 21 — calculating the norm of the 2nd row; 11) a− 77 — calculating the norm of the 7th column; 12) a+ 74 — from the orthogonality condition of the 6th and the 7th rows; 13) a+ 34 — calculating the norm of the 4th column; 14) a− 31 — from the orthogonality condition of the 2nd and the 3rd rows; 15) a+ 11 — calculating the norm of the 1st column; 16) a+ 12 — calculating the norm of the 1st row; 17) a+ 32 — from the orthogonality condition of the 1st and the 2nd columns; 18) a+ 42 — calculating the norm of the 2nd column; 19) a− 35 — calculating the norm of the 3rd row; 20) a+ 45 — from the orthogonality condition of the 3rd and the 4th rows; 21) a+ 46 — calculating the norm of the 4th row; 22) a+ 75 — from the orthogonality condition of the 4th and the 5th columns; 23) a+ 85 — calculating the norm of the 5th column; 24) a+ 86 — from the orthogonality condition of the 5th and the 6th columns; 25) a+ 7,10 — calculating the norm of the 7th column; 26) a− 8,10 — from the orthogonality condition of the 7th and the 8th rows; 27) \|ac 96\| — calculating the norm of the 6th column; 28) a+ 10,10 — calculating the norm of the 10th column; 29) a+ 10,11 — calculating the norm of the 10th row; 30) a+ 8,11 — from the orthogonality condition of the 10th and the 11th columns; 31) a+ 88 — calculating the norm of the 8th row; 32) a+ 9,11 — calculating the norm of the 11th column; 33) \|ac 98\| — calculating the norm of the 9th row; 34) arg ac 96 and arg ac 98 — from the orthogonality condition of the 8th and the 9th rows. Jo ur na l A lg eb ra D is cr et e M at h. S. A. Kruglyak, I. V. Livinsky 227 The next step should be the description of the formulas of the matrix elements and the range of values of independent parameters (expressions inside various radicals should be positive). However, the explicit formulas become very bulky and the description of the range of values of the parameters becomes very difficult. We simplify the problem by letting p = 1 and restricting the range for t. We show that the range of values for t contains an interval. Anyway, this implies that the range of values for t is infinite. As a result, the following statement is proved. Theorem 1. The problem of unitary classification of orthoscalar repre- sentations of the partially ordered set (N, 4) is of infinite type. Proof. Let p = 1, and find consecutively the expressions for all matrix elements via parameter t. a+ 59 = √ 3, a+ 53 = √ t, a+ 5,12 = √ 2− t, a+ 6,12 = √ t + 1, a− 63 = − √ (2−t)(1+t) t , a+ 23 = √ 3t−2 t , a+ 67 = 1, a+ 64 = √ 2(t−1) t , a+ 24 = √ 2(t2−1)(2−t) t(3t−2) , a+ 21 = √ 4−4t+2t2 3t−2 , a− 77 = −1, a+ 74 = √ t 2(t−1) , a+ 34 = √ t(4t2−3t−2) 2(t−1)(3t−2) , a− 31 = − √ (2−t)(1+t)(4t2−3t−2) (3t−2)(4−4t+2t2) , a+ 11 = √ 14−19t+7t2 4−4t+2t2 , a+ 12 = √ 5(2−t)(t−1) 4−4t+2t2 , a+ 32 = √ 5(t−1)(3t−2)(14−19t+7t2) (1+t)(4−4t+2t2)(4t2−3t−2) , a+ 42 = √ (9t−11)(4−4t+2t2) (1+t)(4t2−3t−2) , a− 35 = − √ (3t−2)(8t3−41t2+71t−40) 2(t2−1)(4t2−3t−2) , a+ 45 = √ 10(t−1)2(9t−11)(14−19t+7t2) (1+t)(4t2−3t−2)(8t3−41t2+71t−40) , a− 46 = − √ (5−3t)(4t2−3t−2) 8t3−41t2+71t−40 , a+ 75 = √ 8t3−41t2+71t−40 2(t2−1) , a+ 85 = √ 2(t−1)(−16t4+136t3−455t2+658t−335) (1+t)(8t3−41t2+71t−40) , a+ 86 = √ 5(t−1)(5−3t)(9t−11)(14−19t+7t2) (8t3−41t2+71t−40)(−16t4+136t3−455t2+658t−335) , a+ 7,10 = √ 4(4−t)(t−1) t+1 , a− 8,10 = − √ −16t4+136t3−455t2+658t−335 4(4−t)(t2−1) , ac 96 = √ (29−11t)(8t3−41t2+71t−40) −16t4+136t3−455t2+658t−335 eix, Jo ur na l A lg eb ra D is cr et e M at h. 228 Orthoscalar representations of the set (N, 4) a+ 10,10 = √ 31−37t+12t2 4(4−t)(t−1) , a+ 8,11 = √ (t+1)(31−37t+12t2)(−47+57t−16t2) 4(4−t)(t−1)(−16t4+136t3−455t2+658t−335) , a+ 88 = √ (3−t)(t−1)(89−87t+24t2) −16t4+136t3−455t2+658t−335 , a+ 9,11 = √ 4(4−t)(t2−1)(7−4t) −16t4+136t3−455t2+658t−335 , ac 98 = √ −8t4+89t3−401t2+699t−403 −16t4+136t3−455t2+658t−335 eiy, a+ 10,11 = √ −47+57t−16t2 4(4−t)(t−1) . We are not trying to find exact values of the range for t. Anyway, it is not difficult to show that all the expressions inside radicals are positive for t ∈ [ 7 5 , 8 5 ] . Real numbers x and y can be found from the orthogonality condition of the 10th and the 11th rows. It could be verified straight-forward that the endomorphism ring of the representation Γ (and of the representation T ) is trivial; therefore, the representations Γ and T are indecomposable. Thus, for t ∈ [ 7 5 , 8 5 ] we have two (complex-conjugate) orthoscalar representations of the poset (N, 4). References [1] Ostrovskyi V., Samoilenko Yu., Introduction to the theory of representations of finitely presented ∗-algebras. I. Representations by bounded operators, vol. 11,// Rev. Math. and Phys. — 1999. — 11, N 1. — 261 p. [2] S. A. Kruglyak, V. I. Rabanovich, and Yu. S. Samoilenko, On sums of projectors// Funkts. Anal. Prilozhen., 36, Issue 3, 20 – 35 (2002). [3] S. Albeverio, V. Ostrovskyi, Yu. 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Popovich, Yu. S. Samoilenko, Representations of ∗-algebras associated with Dynkin graphs and Horn’s problem// Uchen. Zap. Tavr. Nats. Univ., 16(55), 133 – 139, (2003). [16] A. V. Roiter, Boxes with an involution. — In book: Representations and quadratic forms. K., 124 – 126 (1979). [17] Yusenko K., Samoilenko Yu., Kleiner’s theorem for unitary representations of posets. — arXiv: 1103.1085 v 2 [math RT] 18 Feb 2012. [18] Futorny V., Samoilenko Yu., Yusenko K., Representations of posets: linear versus unitary. — Journal of Physics — Conference Seriees, ISSN: 1742-6588, Volume: 346, Issue: I/ Page 012006. Contact information S. A. Kruglyak Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 3 Tereshchenkivska st., 01601, Kyiv, Ukraine E-Mail: krug@ehl.kiev.ua I. V. Livinsky University of Toronto, 40 St. George St., Toronto, Ontario, M5S2E4, CANADA E-Mail: ivan.livinskyi@mail.utoronto.ca Received by the editors: 14.05.2012 and in final form 25.05.2012.

Orthoscalar representations of the partially ordered set (N, 4)

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