Alternating Sign Matrices and Latin Squares Cian O'Brien Rachel Quinlan and Kevin Jennings National University of Ireland, Galway [email protected] Postgraduate Modelling Group, NUI Galway October 4th, 2019 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 1 / 9 20 1 0 03 61 −1 1 07 6 7 40 1 −1 15 0 0 1 0 They are a generalisation of the permutation matrices. 2+ 0 0 0 3 6 0 0 0 +7 6 7 4 0 + 0 0 5 0 0 + 0 Alternating Sign Matrices An Alternating Sign Matrix (ASM) is (0; 1; −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign. Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 They are a generalisation of the permutation matrices. 2+ 0 0 0 3 6 0 0 0 +7 6 7 4 0 + 0 0 5 0 0 + 0 Alternating Sign Matrices An Alternating Sign Matrix (ASM) is (0; 1; −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign. 20 1 0 03 61 −1 1 07 6 7 40 1 −1 15 0 0 1 0 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 They are a generalisation of the permutation matrices. 2+ 0 0 0 3 6 0 0 0 +7 6 7 4 0 + 0 0 5 0 0 + 0 Alternating Sign Matrices An Alternating Sign Matrix (ASM) is (0; 1; −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign. 2 0 + 0 0 3 6+ − + 0 7 6 7 4 0 + − +5 0 0 + 0 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 2+ 0 0 0 3 6 0 0 0 +7 6 7 4 0 + 0 0 5 0 0 + 0 Alternating Sign Matrices An Alternating Sign Matrix (ASM) is (0; 1; −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign. 2 0 + 0 0 3 6+ − + 0 7 6 7 4 0 + − +5 0 0 + 0 They are a generalisation of the permutation matrices. Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 Alternating Sign Matrices An Alternating Sign Matrix (ASM) is (0; 1; −1)-matrix for which all row and column sums are 1, and the non-zero elements in each row and column alternate in sign. 2 0 + 0 0 3 6+ − + 0 7 6 7 4 0 + − +5 0 0 + 0 They are a generalisation of the permutation matrices. 2+ 0 0 0 3 6 0 0 0 +7 6 7 4 0 + 0 0 5 0 0 + 0 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 2 / 9 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 + 0 5 % 4+ − +5 % 4 0 + 0 5 0 0 + 0 + 0 + 0 0 Alternating Sign Hypermatrices An Alternating Sign Hypermatrix (ASHM) is a hypermatrix (n × n × n array) for which every plane is an ASM. Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 3 / 9 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 + 0 5 % 4+ − +5 % 4 0 + 0 5 0 0 + 0 + 0 + 0 0 Alternating Sign Hypermatrices An Alternating Sign Hypermatrix (ASHM) is a hypermatrix (n × n × n array) for which every plane is an ASM. Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 3 / 9 Alternating Sign Hypermatrices An Alternating Sign Hypermatrix (ASHM) is a hypermatrix (n × n × n array) for which every plane is an ASM. 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 + 0 5 % 4+ − +5 % 4 0 + 0 5 0 0 + 0 + 0 + 0 0 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 3 / 9 2 0 0 + 0 0 3 6 0 + − + 0 7 6 7 6+ − + − +7 6 7 4 0 + − + 0 5 0 0 + 0 0 Non-Zero Entry Bounds for ASMs The number of non-zero entries in the rows/columns of an ASM is bounded above by (1; 3; 5;:::; 5; 3; 1). Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 4 / 9 Non-Zero Entry Bounds for ASMs The number of non-zero entries in the rows/columns of an ASM is bounded above by (1; 3; 5;:::; 5; 3; 1). 2 0 0 + 0 0 3 6 0 + − + 0 7 6 7 6+ − + − +7 6 7 4 0 + − + 0 5 0 0 + 0 0 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 4 / 9 + 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 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 + 0 0 0 + 0 0 0 + 0 0 0 0 Non-Zero Entry Bounds for ASHMs The number of non-zero entries in the planes of an ASHM is bounded above by 21 1 ··· 1 13 61 3 ··· 3 17 6 7 6. .. .7 6. .7 6 7 41 3 ··· 3 15 1 1 ··· 1 1 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 5 / 9 Non-Zero Entry Bounds for ASHMs The number of non-zero entries in the planes of an ASHM is bounded above by 21 1 ··· 1 13 61 3 ··· 3 17 6 7 6. .. .7 6. .7 6 7 41 3 ··· 3 15 1 1 ··· 1 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 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 + 0 0 0 + 0 0 0 + 0 0 0 0 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 5 / 9 21 2 33 42 3 15 3 1 2 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i;j;k = k=1 k × Mi;j;k . 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 0 +5 % 4+ 0 0 5 % 4 0 + 0 5 0 + 0 0 0 + + 0 0 Latin Squares An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column. Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i;j;k = k=1 k × Mi;j;k . 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 0 +5 % 4+ 0 0 5 % 4 0 + 0 5 0 + 0 0 0 + + 0 0 Latin Squares An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column. 21 2 33 42 3 15 3 1 2 Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i;j;k = k=1 k × Mi;j;k . 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 0 +5 % 4+ 0 0 5 % 4 0 + 0 5 0 + 0 0 0 + + 0 0 Latin Squares An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column. 21 2 33 42 3 15 3 1 2 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices. Cian O'Brien (NUIG) ASBG-Colourings October 4th, 2019 6 / 9 Therefore each latin square corresponds uniquely to a permutation hypermatrix. For a permutation hypermatrix M, define L(M) to be Pn L(M)i;j;k = k=1 k × Mi;j;k . 2+ 0 0 3 2 0 + 0 3 2 0 0 +3 4 0 0 +5 % 4+ 0 0 5 % 4 0 + 0 5 0 + 0 0 0 + + 0 0 Latin Squares An n × n Latin Square is an n × n array of n symbols such that each symbol occurs exactly once in each row and column. 21 2 33 21 0 03 20 1 03 20 0 13 42 3 15 = 40 0 15 + 2 41 0 05 + 3 40 1 05 3 1 2 0 1 0 0 0 1 1 0 0 Each n × n latin square can be decomposed uniquely into a sum of scalar multiples of mutually orthogonal permutation matrices.