Latin square determinants and permanents Ken Johnson Penn State Abington College (Joint work with D. Donovan and I. Wanless)
Outline ◮ 1)The basic idea ◮ 2) Motivation ◮ 3) Summary of the group case ◮ 4) Results on latin square determinants ◮ 5) Ditto for permanents ◮ 6) Questions and further ideas
Latin square matrices Let L be an n × n latin square on the elements { 1 , 2 , ..., n } Let { x 1 , x 2 , ..., x n } be commuting variables. For each i replace i by x i in the square wherever it occurs. This is the latin square matrix Its determinant Θ L is the Latin square determinant of L . Example 1 2 3 . L = 2 3 1 3 1 2 − ( x 3 1 + x 3 2 + x 3 Θ L = 3 − 3 x 1 x 2 x 3 ) − ( x 1 + x 2 + x 3 )( x 1 + ω x 2 + ω 2 x 3 )( x 1 + ω 2 x 2 + ω x 3 ) , = 2 π i 3 . In general Θ L is a homogeneous polynomial of where ω = e degree n in n variables.
Motivation The group determinant of a finite group G = { g 1 , g 2 , ..., g n } is the determinant of the latin square L G whose ( i , j ) th element is x g i g − 1 . j If G is the cyclic group of order 3, then 1 3 2 L G = 2 1 3 3 2 1 and Θ G = Θ L G is the negative of the previous polynomial. Group representation theory began when Frobenius invented a way to factorise Θ G for all G (in principle). In practice, the factorisation of Θ G where G is a cyclic group is more or less equivalent to the fast Fourier transform.
Summary of results in the group case 1) The irreducible factors of Θ G are in 1 : 1 correspondence with the irreducible representations. 2) Suppose the polynomial f corresponds to the representation ρ . Then the character of ρ can be read off from the coefficient of x n − 1 in f . 1 3) G is uniquely determined up to isomorphism by Θ G . 4) G is uniquely determined by the coefficients of x n − 1 , x n − 2 and 1 1 x n − 3 in Θ G . 1 (these coefficients are polynomials). There are connections with random walks on groups, harmonic analysis, ...
First results on latin square determinants They were first discussed in a paper by KWJ in 1986 (proc. Montreal conference) The motivation was to extend the (combinatorial) character theory for quasigroups developed with Jonathan Smith. Result (Peter Cameron) Almost all quasigroups have a trivial character theory, i.e the P -matrix of the corresponding association scheme is � 1 � n − 1 . 1 − 1 If Q is a quasigroup Q of order n and then corresponding to each basic (i.e. irreducible character) ρ of Q there is a factor ψ of Θ Q . For a group ψ is the m th power of the the irreducible polynomial which corresponds to ρ , where m is the degree of ρ . For an arbitrary quasigroup, the polynomial Θ Q is not so nice.
The connection with graph theory The latin square matrix can be interpreted as a transition matrix for a random walk on a loop or quasigroup, where the variables are interpreted as probabilities. If the probabilities are non-zero on a generating set, this is a random walk on the Cayley graph.
Factorisation There is always the trivial factor � n i =1 x i . Example: The Octonion loop O of order 16. The character theory gives eight linear characters (from the homomorphism to the abelian group C 2 × C 2 × C 2 ) and then one factor of degree 8. This factor decomposes into the 4 th power of a sum of eight squares of terms ( x e i − x − e i ) and therefore can be decomposed into linear factors over the Octonions The smallest Moufang loop of order 12 has four linear characters which give rise to linear factors, the remaining factor of degree 8 √ corresponding to the non-linear character (of degree 8) This factor decomposes into two factors of degree 4.
Results by other authors related to determinants and permanents A transversal in a latin square is a set of n entries in distinct rows and columns which are all distinct. (example of C 3 above) A latin square has an orthogonal mate if and only if it has n non-intersecting transversals Balasubramanian (1990) Relates the numbers of transversals to the permanent. Akhbari and Alireza (2004) Transversals and multicolored matchings. Falcon (2008) Determinants of latin squares of a given pattern
First results (continued) A group is determined up to isomorphism by its determinant. For a latin square, define the trisotopy relation T on the set of n × n latin squares by L T M if L is either isotopic to M or to the transpose of M . Suppose f ( x 1 , x 2 , ..., x n ) and g ( x 1 , x 2 , ..., x n ) are polynomials. They are equivalent if there exists a permutation σ ∈ S n such that g ( x 1 , x 2 , ..., x n ) = ± f ( x σ (1) , x σ (2) , ..., x σ ( n ) ). It is clear that trisotopic squares have equivalent determinants. By direct calculation, squares up to order 7 are determined up to trisotopy by their determinants.
However, D. Ford and KWJ calculated that of the 842227 trisotopy classes of squares of size 8 all except 37 are determined by their determinants, and these merge into 12 classes with distinct determinants. Each of the exceptional squares has the form � A � B C D where A and D are 4 × 4 latin squares on { 1 , 2 , 3 , 4 } and B and C are 4 × 4 latin squares on { 5 , 6 , 7 , 8 } . (Among other things their character theory is non-trivial)
Recent results with Donovan-Wanless (on both det( L ) and per( L )) per( L ) is obtained by using the permanent instead of the determinant above. Numerical result: Let L be an n × n latin square. If the monomial terms in either det( L ) or per( L ) are collected, there cannot be more than � 2 n − 1 � − n ( n − 1) n such terms. The permanent does distinguish all the trisotopy squares of order 8. For all m ≥ 3 there exist pairs ( A , B ) of nonisotropic latin squares of order 3 m with equivalent permanents and determinants. If per( A )=per( B ) then A and B have the same number of transversals. (not true for determinants). If per( A )=per( B ) then A and B need not have the same number of orthogonal mates
Numbers of terms An example of the data produced The number of monomial terms (after collection) for the squares of order 8 can be summarised as follows: For the circulant (cyclic group) 810 in both det and per. mean in det 5174 mean in per 5759 most in det 5491 most in per 6054 bound calculated earlier 6379 Results suggest that the minimum should always occur at the circulant.
Questions ◮ Are there squares with equal permanents but dissimilar determinants? ◮ Are there pairs with equal determinants, but with autotopism groups of different orders? ◮ The permanent or determinant determines the number of 2 × 2 subsquares. Can squares with equal permanents have different numbers of k × k subsquares? ◮ Is there a square which is not isotopic to a group matrix, but which has the same permanent (or determinant) as that of a group? ◮ Among squares of order n , does the circulant minimise the number of monomials in the determinant or permanent?
Dickson’s results on the mod p case The group determinant mod p of a p -group. Lemma Let H be any p-group of order r = p s . Let P be the upper triangular matrix of the form 1 1 1 1 ... 1 1 2 3 r − 1 1 3 ( r − 1)( r − 2) / 2 . 1 ... ... r − 1 1 Then a suitable ordering of H exists such that, modulo p, PX H P − 1 is a lower triangular matrix with identical diagonal entries of the form α = � r i =1 x h i . The group determinant Θ H modulo p is thus α r .
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