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Generalized Alternating Sign Matrices and Signed Permutation Matrices

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Generalized Alternating Sign Matrices and Signed Permutation Matrices J. Korean Math. Soc. 58 (2021), No. 4, pp. 921{948 https://doi.org/10.4134/JKMS.j200318 pISSN: 0304-9914 / eISSN: 2234-3008 GENERALIZED ALTERNATING SIGN MATRICES AND SIGNED PERMUTATION MATRICES Richard A. Brualdi and Hwa Kyung Kim Abstract. We continue the investigations in [6] extending the Bruhat order on n × n alternating sign matrices to our more general setting. We show that the resulting partially ordered set is a graded lattice with a well-define rank function. Many illustrative examples are given. 1. Introduction An n × n alternating sign matrix (abbreviated to ASM) is a (0; ±1) matrix such that, ignoring 0's, the +1's and −1's in each row and each column alternate beginning and ending with +1. The origins and many properties of ASMs can be found in [1,3,9{11]. In [6] a generalization of ASMs was defined as follows: Let u = (u1; u2;:::, 0 0 0 0 0 0 0 0 un), u = (u1; u2; : : : ; un), v = (v1; v2; : : : ; vm), and v = (v1; v2; : : : ; vm) be vectors of ±1's. If A is an m × n matrix, we define A(u; u0jv; v0) to be the (m + 2) × (n + 2) matrix (1) with rows indexed by 0; 1; : : : ; m + 1 and columns indexed by 0; 1; : : : ; n + 1. 0 u1 u2 ··· un−1 un 0 0 v1 v1 0 v2 v2 0 0 . (1) A(u; u jv; v ) = . A . : 0 vm−1 vm−1 0 vm vm 0 0 0 0 0 u1 u2 ··· un−1 un 0 We then write A = A(u; u0jv; v0)[1; 2; : : : ; mj1; 2; : : : ; n] to denote that A is the middle m × n submatrix of A(u; u0jv; v0), and also write A(u; u0jv; v0) ! A. Received June 4, 2020; Revised August 21, 2020; Accepted August 31, 2020. 2010 Mathematics Subject Classification. 05B20, 15B48. Key words and phrases. Permutation matrix, alternating sign matrix, Bruhat order. The second author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2016R1D1A1B03930279). c 2021 Korean Mathematical Society 921 922 R. A. BRUALDI AND H. K. KIM A(u; u0jv; v0)-ASM is an m×n (0; ±1)-matrix A as in (1) such that, ignoring 0's, the +1's and −1's in rows 0; 1; 2; : : : ; m + 1 and columns 0; 1; 2; : : : ; n + 1 of the (0; ±1)-matrix A(u; u0jv; v0) alternate. Thus the condition that the first and last nonzero entry in each row and column of an ASM is +1 is relaxed, where now the first and last nonzero of each row is determined by v and v0, respectively, and the first and last nonzero of each column is determined by u and u0, respectively. Note also that, unlike for ASMs, the first and last rows and columns of a (u; u0jv; v0)-ASM depend on u; u0; v; v0 and so may con- tain more than one nonzero entry. We denote the set of (u; u0jv; v0)-ASMs 0 0 0 0 by Am;n(u; u jv; v ). If u = u and v = v and m = n, we often abbrevi- ate these notations to: (u; v)-ASM and An(u; v), respectively. If v = u, we also use the abbreviations (u)-ASM and An(u). Observe that if m = n and u = v = (−1; −1;:::; −1), then a (u)-ASM is an ordinary ASM, and An(u) is the usual set An of n × n ASMs. Example 1.1. Let m = 2 and n = 3, and let u = (1; −1; 1), u0 = (−1; 1; −1), v = (1; −1) and v = (1; −1). Then a (u; u0jv; v0)-ASM is given below: 2 0 1 −1 1 0 3 6 1 −1 1 −1 1 7 −1 1 −1 6 7 ! : 6 −1 1 −1 1 −1 7 1 −1 1 4 5 0 −1 1 −1 0 In [6], necessary and sufficient conditions are given for the nonemptiness of 0 0 Am;n(u; u jv; v ). In this paper we primarily consider the case where m = n 0 0 and u = u and v = v , that is, the set An(u; v). In this case, a 180 degree rotation gives a bijection from An(u; v) to An(u; v) where u = (un; : : : ; u1) and v = (vn; : : : ; v1). We also sometimes restrict our attention to the case where u = v, as in ordinary ASMs; in this case, a 90 degree clockwise rotation give a bijection from An(u) to An(u). Also An(u) is invariant under a reflection over the main diagonal. The nonemptiness of An(u; v) is easily decided. Lemma 1.2. Let u = (u1; u2; : : : ; un) and v = (v1; v2; : : : ; vn) be vectors of ±1's. Then An(u; v) 6= ; if and only if u and v contain the same number of +1's. Proof. Suppose that u contains p (+1)'s and (n − p)(−1)'s and v contains q (+1)'s and (n − q)(−1)'s. Let A 2 An(u; v). Then p columns of A sum to −1 and (n − p) columns sum to +1. Similarly, q rows of A sum to −1 and (n − q) rows sum to +1. Hence −p + (n − p) = −q + (n − q) implying that p = q. Conversely, if p = q, let Iu;v be the (0; ±1)-matrix whose p × p submatrix in those columns for which ui = +1 and those rows for which vi = +1 equals −Ip, and whose (n − p) × (n − p) submatrix in those columns for which ui = −1 GENERALIZED ALTERNATING SIGN MATRICES 923 and those rows for which vi = −1 equals In−p, with all other entries equal to 0. Thus Iu;v is in An(u; v). Note that in the above proof, we can replace Ip by any p × p permutation matrix and In−p by any (n − p) × (n − p) permutation matrix, and these p!(n − p)! signed permutation matrices (definition in the next example) are the only signed permutation matrices in An(u; v). Example 1.3. Here and elsewhere we usually use `+' in place of +1 and `−' in place of −1. Also, when we block the cells of a matrix, we usually do not put in 0's with an empty cell signifying a 0. Let u = (−; −; +; +; −) and v = (−; +; −; +; −). Then the following matrix A is in An(u; v). 0 − − + + − 0 − + − 2 + 3 + − + 6 − 7 6 7 − + − ! A = 6 + 7 : 6 7 + − + 4 − 5 − + − + 0 − − + + − 0 The matrix A has a special property in that there is exactly one nonzero in each row and in each column, as in the proof of Lemma 1.2. Thus A is a signed permutation matrix, that is, a permutation matrix in which some of the 1's have been replaced with −1's. In order that An(u; v) contain a signed permutation matrix it is necessary and sufficient, as in this example, that u and v have the same number p of +1's and so the same number (n − p) of −1's. Let u and v be vectors of ±1's each containing p (+1)'s and (n − p)(−1)'s. Let A 2 An(u; v). Then the +1's determine a p × p submatrix A+ of A and the −1's determine the complementary (n−p)×(n−p) submatrix A− of A. Let A+ be −P where P is a p×p permutation matrix, and let A− be the (n−p)×(n−p) matrix Q where Q is a permutation matrix, and where all other entries of A equal 0. Then A is a signed permutation matrix in An(u; v), and every signed permutation matrix in An(u; v) arises in this way. Let Lp be the p × p anti- identity (0; 1) matrix (1's on the antidiagonal). If P = Lp (so A+ = −Lp) and Q = In−p (so A− = In−p), then the resulting signed permutation matrix is denoted by In(u; v) and is called the (u; v)-identity matrix. If P = Ip (so A+ = −Ip) and Q = Ln−p (so A− = Ln−p) matrix, then the resulting signed permutation matrix is denoted by Ln(u; v) and is called the (u; v)-anti-identity matrix. 924 R. A. BRUALDI AND H. K. KIM Example 1.4. Continuing with Example 1.3, we get the (u; v)-identity matrix as given in 0 − − + + − 0 − + − 2 + 3 + − + 6 − 7 6 7 − + − ! 6 + 7 = I5(u; v); 6 7 + − + 4 − 5 − + − + 0 − − + + − 0 and the (u; v)-anti-identity matrix as given in 0 − − + + − 0 − + − 2 + 3 + − + 6 − 7 6 7 − + − ! 6 + 7 = L5(u; v): 6 7 + − + 4 − 5 − + − + 0 − − + + − 0 For any m × n matrix A = [aij], the sum-matrix (also called corner matrix in [11]) Σ(A) = [σij] of A is the m × n matrix where X X σij = σij(A) = akl (1 ≤ i ≤ m; 1 ≤ j ≤ n); k≤i l≤j the sum of the entries in the leading i × j submatrix of A. Define σij = 0 if i = 0 or j = 0. Then the matrix A is uniquely determined by its sum-matrix Σ(A), namely, aij = σij − σi;j−1 − σi−1;j + σi−1;j−1 (1 ≤ i; j ≤ n): The sum-matrix Σ(A) = [σij] of a matrix A 2 An(u; v) has the following properties which are easily verified: • The entries in row i and column i of Σ(A) are taken from the set f0; ±1; ±2;:::; ±ig.
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