Introduction Operator Induction & Reduction Operator Calculus Applications Combinatorial semigroups and induced/deduced operators G. Stacey Staples Department of Mathematics and Statistics Southern Illinois University Edwardsville
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Hypercube Q 4
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Multi-index notation Let [ n ] = { 1 , 2 , . . . , n } and denote arbitrary, canonically ordered subsets of [ n ] by capital Roman characters. 2 [ n ] denotes the power set of [ n ] . Elements indexed by subsets: � γ J = γ j . j ∈ J Natural binary representation
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Modified hypercubes
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Modified hypercubes
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Special elements γ ∅ (identity) γ α (commutes with generators, γ α 2 = γ ∅ ) 0 γ (“absorbing element” or “zero” ) “Special” elements do not contribute to Hamming weight.
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Groups Nonabelian – B p , q “Blade group” (Clifford Lipschitz groups) γ i γ j = γ α γ j γ i ( 1 ≤ i � = j ≤ p + q ) � γ ∅ 1 ≤ i ≤ p , 2 = γ i γ α p + 1 ≤ i ≤ p + q Abelian – B p , q sum “Abelian blade group” γ i γ j = γ j γ i ( 1 ≤ i � = j ≤ p + q ) � γ ∅ 1 ≤ i ≤ p , 2 = γ i γ α p + 1 ≤ i ≤ p + q
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Semigroups Nonabelian – “Null blade semigroup” Z n γ i γ j = γ α γ j γ i ( 1 ≤ i � = j ≤ n ) � 0 1 ≤ i ≤ n , 2 = γ i γ ∅ i = α Abelian – “Zeon semigroup” Z n sym γ i γ j = γ j γ i ( 1 ≤ i � = j ≤ n ) � 0 1 ≤ i ≤ n , 2 = γ i γ ∅ i = ∅
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Passing to semigroup algebra: Canonical expansion of arbitrary u ∈ A : � = u J γ J u J ∈ 2 [ n ] ∪{ α } + γ J + γ α − γ J . � � = u J u J J ∈ 2 [ n ] J ∈ 2 [ n ] Naturally graded by Hamming weight (cardinality of J ).
Introduction Operator Induction & Reduction Modified Hypercubes Operator Calculus Particular groups & semigroups Applications Group or Quotient Isomorphic Semigroup Algebra Algebra B p , q R B p , q / � γ α + γ ∅ � C ℓ p , q B p , q sym R B p , q sym / � γ α + γ ∅ � C ℓ p , q sym � R n Z n R Z n / � 0 γ , γ α + γ ∅ � Z n sym R Z n sym / � 0 γ � C ℓ n nil
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Idea: Induced Operators Let V be the vector space spanned by generators { γ j } of 1 (semi)group S . Let A be a linear operator on V . 2 A naturally induces an operator A on the semigroup 3 algebra R S according to action (multiplication, conjugation, etc.) on S . � A ( γ J ) := A ( γ j ) j ∈ J
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators The Clifford algebra C ℓ p , q Real, associative algebra of dimension 2 n . 1 Generators { e i : 1 ≤ i ≤ n } along with the unit scalar 2 e ∅ = 1 ∈ R . Generators satisfy: 3 [ e i , e j ] := e i e j + e j e i = 0 for 1 ≤ i � = j ≤ n , � 1 if 1 ≤ i ≤ p , 2 = e i − 1 if p + 1 ≤ i ≤ p + q .
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Rotations & Reflections: x �→ uvxvu
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Hyperplane Reflections Product of orthogonal vectors is a blade . 1 Given unit blade u ∈ C ℓ Q ( V ) , where Q is positive definite. 2 The map x �→ u x u − 1 represents a composition of 3 hyperplane reflections across pairwise-orthogonal hyperplanes. This is group action by conjugation. 4 Each vertex of the hypercube underlying the Cayley graph 5 corresponds to a hyperplane arrangement.
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Blade conjugation u ∈ B ℓ p , q ≃ C ℓ Q ( V ) a blade. 1 Φ u ( x ) := u x u − 1 is a Q -orthogonal transformation on V . 2 Φ u induces ϕ u on C ℓ Q ( V ) . 3 The operators are self-adjoint w.r.t. �· , ·� Q ; i.e., they are 4 quantum random variables . Characteristic polynomial of Φ u generates Kravchuk 5 polynomials .
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Blade conjugation Conjugation operators allow factoring of blades. 1 Eigenvalues ± 1 Basis for each eigenspace provides factorization of corresponding blade. Quantum random variables obtained at every level of 2 induced operators. ϕ ( ℓ ) is self-adjoint w.r.t. Q -inner product for each ℓ = 1 , . . . , n . Kravchuk polynomials appear in traces at every level. 3 Kravchuk matrices represent blade conjugation operators 4 (in most cases 1 ). 1 G.S. Staples, Kravchuk Polynomials & Induced/Reduced Operators on Clifford Algebras , Preprint (2013).
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators More generally... Suppose X is a linear operator on V . 1 Suppose I , J ∈ 2 | V | with | I | = | J | = ℓ . 2 Then, � v I | X ( ℓ ) | v J � = det ( X IJ ) . 3 Here, X IJ is the submatrix of X formed from the rows indexed by I and the columns indexed by J . This holds for C ℓ Q ( V ) as well as � V . In the latter case, X is block diagonal.
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators nil The zeon algebra C ℓ n Real, associative algebra of dimension 2 n . 1 Generators { ζ i : 1 ≤ i ≤ n } along with the unit scalar 2 ζ ∅ = 1 ∈ R . Generators satisfy: 3 [ ζ i , ζ j ] := ζ i ζ j − ζ j ζ i = 0 for 1 ≤ i , j ≤ n , ζ i ζ j = 0 ⇔ i = j .
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Zeons Applications in combinatorics, graph theory, quantum 1 probability explored in monograph by Schott & Staples 2 . Based on papers by Staples and joint work with Schott. Induced maps appear in work by Feinsilver & McSorley 3 2 2 Operator Calculus on Graphs (Theory and Applications in Computer Science) , Imperial College Press, London, 2012 3 P . Feinsilver, J. McSorley, Zeons, permanents, the Johnson scheme, and generalized derangements, International Journal of Combinatorics , vol. 2011, Article ID 539030, 29 pages, 2011. doi:10.1155/2011/539030
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Adjacency matrices Let G = ( V , E ) be a graph on n vertices. 1 Let A denote the adjacency matrix of G , viewed as a linear 2 transformation on the vector space generated by V = { v 1 , . . . , v n } . A ( k ) denotes the multiplication-induced operator on the 3 grade- k subspace of the semigroup algebra C ℓ V nil .
Introduction Induced Operators Operator Induction & Reduction * Operators on Clifford algebras Operator Calculus * Operators on zeons Applications Reduced / Deduced Operators Theorem For fixed subset I ⊆ V, let X I denote the number of disjoint cycle covers of the subgraph induced by I. Similarly, let M J denote the number of perfect matchings on the subgraph induced by J ⊆ V (nonzero only for J of even cardinality). Then, � � tr ( A ( k ) ) = X I \ J M J . J ⊆ I I ⊂ V | I | = k
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