{
  "paper_id": "oai:elib.bsu.by:123456789/347370",
  "source_pages": 3,
  "license_notice": "Translated from Russian. Original: Козел, П. Т., “О строении изометрических преобразований пространства с симплектической метрикой”, Vestnik of the V. I. Lenin Belarusian State University. Series 1: Physics. Mathematics. Mechanics, 1978, no. 1, pp. 3–5. Published by Belarusian State University; license CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/) as stated in the item's record in the BSU Electronic Library. Changes: translated into English.",
  "title_en": "On the structure of isometric transformations of a space with a symplectic metric",
  "title_ru": "О строении изометрических преобразований пространства с симплектической метрикой",
  "authors": [
    "P. T. Kozel"
  ],
  "abstract_en": "Let V be a nondegenerate 2l-dimensional symplectic space over the finite field F_q and let σ be an isometry of V whose characteristic polynomial p(x) is irreducible over F_q, so that p is self-reciprocal. The note proves that V has a basis with Gram matrix [[0, aE_l], [-aE_l, 0]], a ≠ 0, in which σ is represented by an explicit companion-type matrix built from the coefficients of p(x). The basis is constructed from a nonzero solution N_1 of the quadratic system Xσ^k X = 0, k = 1, …, l-1, and its images under σ.",
  "abstract_source": "generated",
  "translator_notes": [
    {
      "page": 1,
      "note": "The journal's section banner “Математика” (Mathematics) above the paper is kept as an italic line; the running page numbers are omitted."
    },
    {
      "page": 1,
      "note": "Matrix (3) is set in the source with rows of dots; \\hdotsfor{9} renders them. The exponent in the first row of (1) reads x^n in the scan; a^{...} subscripts have been transcribed as printed."
    },
    {
      "page": 2,
      "note": "Russian typesetting repeats the operator sign at the start of a continued line of a display (for example “−” in (6) and in the long computation of N_j M_i); each sign is written once here."
    },
    {
      "page": 2,
      "note": "In the final chain for N_j M_i the source writes σ^{l+j−i} and σ^{j−1} as printed; the middle line contains an unclosed parenthesis in the source, closed here as the sense requires."
    },
    {
      "page": 3,
      "note": "Page 3 also carries the beginning of the next article (V. P. Kirlitsa, A. F. Gomez, “A maximum principle for systems with delay in the control”, UDC 62-50); it is not part of this paper and is not translated."
    },
    {
      "page": 3,
      "note": "References 1 and 4 are Russian editions of foreign books (E. Artin, Geometric Algebra; J.-P. Serre, Cours d'arithmétique), given with their original titles; reference 3 has no title in the source."
    }
  ],
  "produced_with": "Claude Fable 5.1 (Claude Code), reading the scan rendered to 250 dpi PNG with macOS PDFKit; no OCR.",
  "body_md": "<!-- source-page: 001 -->\n\n*Mathematics*\n\nUDC 512.831\n\nP. T. KOZEL\n\n# ON THE STRUCTURE OF ISOMETRIC TRANSFORMATIONS OF A SPACE WITH A SYMPLECTIC METRIC\n\nLet \\(V\\) be a nondegenerate \\(n\\)-dimensional space with a symplectic metric over the finite field \\(F_q\\) of \\(q\\) elements. The dimension of the space \\(V\\), \\(n = \\dim V = 2l\\), is an even number [1]. Let \\(\\sigma\\) be an isometry of the space \\(V\\) whose characteristic polynomial \\(p(x)\\) is irreducible over the field \\(F_q\\); then, as Zassenhaus showed [2], the polynomial \\(p(x)\\) has the form (see also [3])\n\n\\[\n\\begin{gathered}\np(x) = x^{n} + a_{n-1}x^{n-1} + \\dots + a_{1}x + 1,\\\\\n\\text{where}\\quad a_{n-i} = a_{i},\\quad i = 1, \\dots, n-1,\\quad n = 2l.\n\\end{gathered}\n\\tag{1}\n\\]\n\nThe following holds.\n\n**Theorem.** *Let \\(\\sigma\\) be an isometry of the space \\(V\\) with an irreducible characteristic polynomial of the form (1). Then there exists a basis of the space \\(V\\) with Gram matrix*\n\n\\[\nG = \\begin{bmatrix} 0 & aE_{l} \\\\ -aE_{l} & 0 \\end{bmatrix}, \\qquad 0 \\ne a \\in F_q,\n\\tag{2}\n\\]\n\n*in which the isometry \\(\\sigma\\) corresponds to the matrix*\n\n\\[\nA = \\begin{bmatrix}\n0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & -1 \\\\\n1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & -a_{1} \\\\\n\\hdotsfor{9} \\\\\n0 & 0 & \\dots & 1 & 0 & 0 & \\dots & 0 & -a_{l-1} \\\\\n0 & 0 & \\dots & 0 & 1 & -a_{1} & \\dots & -a_{l-1} & -a_{l} \\\\\n0 & 0 & \\dots & 0 & 0 & 1 & \\dots & 0 & 0 \\\\\n\\hdotsfor{9} \\\\\n0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 1 & 0\n\\end{bmatrix}.\n\\tag{3}\n\\]\n\n**Proof.** Let \\(f\\) be the alternating form defining the metric of the space \\(V\\). For \\(X, Y \\in V\\) we shall denote \\(f(X, Y)\\) by \\(XY\\). Let \\(\\sigma\\) be an isometry of the space \\(V\\) with a single elementary divisor of the form (1). The dimension of the space \\(V\\) is an even number, \\(n = 2l\\) [1]. Consider the system of equations\n\n\\[\nX\\sigma X = 0, \\ \\dots, \\ X\\sigma^{l-1}X = 0, \\qquad X \\in V.\n\\tag{4}\n\\]\n\nLet \\(A_1, \\dots, A_n\\) be some basis of the space \\(V\\). Then the system (4) is a system of homogeneous equations of the second degree in the \\(n\\) unknowns \\(x_1, \\dots, x_n\\), the coordinates of the vector \\(X = x_1A_1 + \\dots + x_nA_n\\). Since the sum of the degrees of the equations of the system (4) equals \\(2(l-1) < n\\), the number of unknowns, the system (4) has a nonzero solution [4, Ch. 1, § 2]. Let\n\n<!-- source-page: 002 -->\n\n\\(N_1\\) be a nonzero vector satisfying the system (4). Consider the collection of vectors\n\n\\[\nN_1,\\ N_2 = \\sigma N_1,\\ \\dots,\\ N_l = \\sigma^{l-1}N_1,\\ M_1 = \\sigma^{l}N_1,\\ \\dots,\\ M_l = \\sigma^{n-1}N_1.\n\\tag{5}\n\\]\n\nWe note some properties of the system (5).\n\n**1.** The system of vectors (5) is linearly independent, i.e., is a basis of the space \\(V\\). Indeed, let \\(m\\) be the smallest number for which \\(\\lambda_0 N_1 + \\lambda_1\\sigma N_1 + \\dots + \\lambda_m\\sigma^m N_1 = 0\\), where not all \\(\\lambda_i = 0\\). Then the polynomial \\(f(x) = \\lambda_0/\\lambda_m + \\lambda_1/\\lambda_m\\, x + \\dots + x^m\\) is the minimal polynomial of the vector \\(N_1\\) with respect to the isometry \\(\\sigma\\). Consequently, \\(f(x)\\) is a divisor of the minimal polynomial of the space \\(V\\), which coincides with \\(p(x)\\) [5, Ch. VII]. Since \\(p(x)\\) is an irreducible polynomial, \\(f(x) = p(x)\\) and \\(m = n\\). Hence follows the independence of the system (5).\n\n**2.** It follows directly from the construction of the system (5) that each vector of the basis (5) is carried by the isometry \\(\\sigma\\) into the next one, with the exception of the last basis vector. To compute \\(\\sigma M_l\\) we use the fact that \\(p(\\sigma) = 0\\). Taking into account the properties of the coefficients of \\(p(x)\\), we obtain\n\n\\[\n\\begin{aligned}\n\\sigma M_l = \\sigma^n N_1 = \\sigma^{2l} N_1 = {} & -N_1 - a_1 N_2 - \\dots - a_{l-1} N_l - a_l M_1 \\\\\n& - a_{l-1} M_2 - \\dots - a_1 M_l.\n\\end{aligned}\n\\tag{6}\n\\]\n\n**3.** Since the vector \\(N_1\\) is a solution of the system (4), \\(N_1\\sigma^m N_1 = 0\\), \\(m = 1, \\dots, l-1\\). Setting \\(\\sigma^0 = 1\\) and taking into account that in the space \\(V\\) every vector is isotropic and that \\(\\sigma^{-1}\\) is also an isometry, we obtain \\(N_1\\sigma^m N_1 = 0\\), \\(m = 0, \\pm 1, \\dots, \\pm(l-1)\\). From this follows also the equality\n\n\\[\n\\sigma^j N_1 \\sigma^{m+j} N_1 = 0, \\qquad m = 0, \\pm 1, \\dots, \\pm(l-1), \\quad j = 1, 2, \\dots\n\\tag{7}\n\\]\n\nFrom (7) it follows that each vector of the sequence (5) is orthogonal to the \\(l-1\\) vectors that immediately follow it. Therefore any \\(l\\) vectors of the sequence (5) that immediately follow one another are pairwise orthogonal. Each vector of the system (5) is also orthogonal to the \\(l-1\\) vectors immediately preceding it. Consider the vectors \\(N_{-i} = \\sigma^{-i}N_1\\), \\(i = l-1, \\dots, 1\\). We shall regard them as preceding the vector \\(N_1\\). Then the properties noted can be written in the form\n\n\\[\n\\begin{gathered}\nN_j N_t = 0, \\quad j = 1, \\dots, l;\\ -(l-j) \\le t \\le l,\\ t \\ne 0;\\\\\nN_j M_t = 0 \\ \\text{if } j > t, \\qquad M_j M_t = 0, \\quad j, t = 1, \\dots, l.\n\\end{gathered}\n\\tag{8}\n\\]\n\n**4.** No system of \\(l+1\\) vectors in the sequence (5) can be pairwise orthogonal, since the maximal dimension of a totally isotropic subspace in the space \\(V\\) equals \\(l = n/2\\) [1]. Consequently, in the system \\(N_1, \\dots, N_l, M_1\\) not all the vectors are pairwise orthogonal. Since any \\(l\\) consecutive vectors in the sequence (5) are pairwise orthogonal, \\(N_1 M_1 = a \\ne 0\\), whence\n\n\\[\nN_j M_j = a, \\qquad j = 1, \\dots, l.\n\\tag{9}\n\\]\n\nFormulas (8) and (9) do not contain the products \\(N_j M_i\\), \\(j < i\\). Let us find these products.\n\n**5.** We shall show that\n\n\\[\nN_j M_i = -a_1 N_j M_{i-1} - \\dots - a_{i-j} N_j M_j \\quad \\text{if } j < i.\n\\tag{10}\n\\]\n\nAccording to the notation of (5), \\(N_j M_i = \\sigma^{j-1}N_1 \\sigma^{l+i-1}N_1\\). Since \\(\\sigma\\) is an isometry, in a product of the form \\(\\sigma^m X \\sigma^k Y\\) the exponents \\(m\\) and \\(k\\) can be increased and decreased by one and the same number. In computing \\(N_j M_i\\) we shall once increase all the exponents by \\(l-i+1\\), in order to use formula (6), and then decrease all the exponents by \\(l-i+1\\); therefore\n\n\\[\n\\begin{aligned}\nN_j M_i &= \\sigma^{l+j-i}N_1 \\cdot \\sigma^{2l}N_1 = \\sigma^{l+j-i}N_1 \\cdot \\bigl(-N_1 - a_1\\sigma N_1 - \\dots \\\\\n&\\quad - a_2\\sigma^{n-2}N_1 - a_1\\sigma^{n-1}N_1\\bigr) = \\sigma^{j-1}N_1\\bigl(-\\sigma^{-(l-i+1)}N_1 \\\\\n&\\quad - a_1\\sigma^{-(l-i)}N_1 - \\dots - a_2\\sigma^{l+i-3}N_1 - a_1\\sigma^{l+i-2}N_1\\bigr) = \\\\\n&= N_j\\bigl(-N_{-(l-i+1)} - a_1 N_{-(l-i)} - \\dots - a_2 M_{i-2} - a_1 M_{i-1}\\bigr).\n\\end{aligned}\n\\]\n\n<!-- source-page: 003 -->\n\nThe bracket is a linear combination of the members of the sequence (\\(j < i\\))\n\n\\[\nN_{-(l-i+1)}, \\ \\dots, \\ N_{-1}, \\ N_1, \\ \\dots, \\ N_l, \\ M_1, \\ \\dots, \\ M_{j-1}, \\ M_j, \\ \\dots, \\ M_{i-1}.\n\\tag{11}\n\\]\n\nThe indices of the vectors \\(N\\) in (11) fall within the bounds of the index \\(t\\) of formulas (8); therefore the product of \\(N_j\\) with each \\(N\\) from (11) equals zero. From (8) it also follows that \\(N_j M_1 = \\dots = N_j \\cdot M_{j-1} = 0\\); therefore in the expression for \\(N_j M_i\\) only a linear combination of the products \\(N_j M_j, \\dots, N_j M_{i-1}\\) remains. It is not hard to see that it will coincide with (10).\n\n**6.** We now construct a basis of the space \\(V\\) with Gram matrix (2). Consider the system of vectors\n\n\\[\n\\begin{gathered}\nN_1, \\ \\dots, \\ N_l, \\quad M'_1 = M_1, \\quad M'_2 = M_2 + a_1 M_1, \\quad M'_3 = {}\\\\\n= M_3 + a_1 M_2 + a_2 M_1, \\ \\dots;\\\\\nM'_l = M_l + a_1 M_{l-1} + a_2 M_{l-2} + \\dots + a_{l-1} M_1,\n\\end{gathered}\n\\tag{12}\n\\]\n\nwhere \\(N_j, M_j\\) are the vectors of the system (5) and \\(a_j\\) are the coefficients of the polynomial \\(p(x)\\).\n\nWe shall show that the Gram matrix of the system (12) coincides with the matrix (2). From (8) and (9) it is easy to obtain that \\(N_j N_t = M'_j M'_t = 0\\), \\(j, t = 1, \\dots, l\\); \\(N_j M'_t = 0\\) if \\(j > t\\); \\(N_j M'_j = a\\), \\(j = 1, \\dots, l\\). Finally, the equality \\(N_j M'_i = 0\\) for \\(j < i\\) follows from (10).\n\n**7.** We shall show that the matrix of the isometry \\(\\sigma\\) in the basis (12) coincides with the matrix (3). We have \\(\\sigma N_i = N_{i+1}\\), \\(i = 1, \\dots, l-1\\), \\(\\sigma N_l = M_1 = M'_1\\), according to the construction of the system (5). \\(\\sigma M'_1 = M_2 = M_2 + a_1 M_1 - a_1 M_1 = M'_2 - a_1 M'_1\\); \\(\\sigma M'_2 = M_3 + a_1 M_2 = M_3 + a_1 M_2 + a_2 M_1 - a_2 M_1 = M'_3 - a_2 M'_1\\); \\(\\dots\\); \\(\\sigma M'_{l-1} = M_l + a_1 M_{l-1} + \\dots + a_{l-2} M_2 + a_{l-1} M_1 - a_{l-1} M_1 = M'_l - a_{l-1} M'_1\\). In computing \\(\\sigma M'_l\\) one should use formula (6). We obtain \\(\\sigma M'_l = -N_1 - a_1 N_2 - \\dots - a_{l-1} N_l - a_l M'_1\\).\n\n**8.** It is verified directly that the matrix (3) has a single elementary divisor \\(p(x)\\) and that \\({}^{T}\\!A\\,G\\,A = G\\). The theorem is proved.\n\n## References\n\n1. E. Artin, Geometric Algebra (Russian translation, Moscow, 1969).\n2. H. Zassenhaus, Can. Math. Bull., 1958, 1, No. 1, 31–39; No. 2, 101–111.\n3. I. K. Tsikumov, Ukr. Mat. Zh., 1966, 18, No. 4.\n4. J.-P. Serre, A Course in Arithmetic (Russian translation, Moscow, 1972).\n5. F. R. Gantmakher, Theory of Matrices (Russian), Moscow, 1954.\n\nReceived by the editors June 30, 1975\n\nDepartment of Algebra\n"
}