Wishart α-determinant, α-hafnian

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1 Wishart α-determinant, α-hafnian (, JST CREST) (, JST CREST), Wishart,. ( )Wishart,. determinant Hafnian analogue., ( )Wishart,. 1 Introduction, Wishart. p ν M = (µ 1,..., µ ν ) = (µ ij ) i=1,...,p p p Σ = (σ i,j ) fix = M tm. j=1,...,ν X 1 = (x i1 ) 1 i p, X 2 = (x i2 ) 1 i p,..., X ν = (x iν ) 1 i p ν p, N p (µ 1, Σ),..., N p (µ ν, Σ),, N p (µ i, Σ) µ i, Σ. X (X 1,..., X p ) ν p, W = (w ij ) W = X tx., W, W Wishart W p (ν, Σ, )., mean square matrix, = 0 Wishart W p (ν, Σ).,, W = X t X Wishart CWp (ν, Σ, ).,. x = (x 1,..., x m ) D. n = (n 1,..., n m ), x n = x n 1 1 xn m m E[x n 1 1 xn m m ] D (n- ). D, D., Wishart, Wishart : E[e tr(θw ) ] = det(i 2ΘΣ) ν 2 e 1 2 tr(i (I 2ΘΣ) 1 )Ω,, Θ p p symmetric parameter matrix, Ω ΣΩ = 1

2 . Wishart ( [2, 11] ). Wishart E[w i1,i 2 ]., determinant analogue. E[w i1,i 2 ]. [5]. 2 Notation of graphs,.,. v w, v w {v, w}. v w, {v, w} = {w, v}., self loop {v, v}. V, U, K V K V,U : K V,U = { {v, u} v V, u U, v u }, K V = K V,V = { {v, u} v u V }. V E K V G = (V, E ), vertex(e ) = { v V {v, u} E for some u V }. (V, K ). E K (V, K ) : {v, u}, {v, u } E = u = u., ( ) 1. M (V, K ) (V, K )., (V, K V ) M (V, K V ) M (V )., (V, K ) E vertex(e ) = V, E perfect. ( 1.) P (V, K ) (V, K ) perfect matchings. P (V ) = P(V, K V ).. v w (v, u). v u, (v, u) (u, v).,, self loop (v, v)., self loop. V, U, K V K V,U K V,U = { (v, u) v V, u U }, K V = K V,V = { (v, u) v, u V }. V E K V G = (V, E), start(e) end(e) 2

3 : start(e) = { v V (v, u) E for some u V }, end(e) = { u V (v, u) E for some v V }. (V, K). 2 E K (V, K) : (v, u), (v, u ) E = u = u (v, u), (v, u) E = v = v., ( ) ( ) 1. (V, K) M(V, K). E M(V, K) start(e) = V end(e) = V, perfect, P(V, K) (V, K) perfect matching. Perfect matching 1., M(V ) = M(V, K V ), P(V ) = P(V, K V ). Remark 2.1. V Z, V = { v v V }, V = { v v V }, l = 2l 1, l = 2l., (V, E) 2 ( V, V, { { v, ü} (v, u) E })., M(V, K) 2 ( V, V, { { v, ü} (v, u) E }). M(V, K). 3 Definition of our polynomials l Z, l = 2l 1, l = 2l. n Z>0 fix, V, V fix : V = [n] = { 1,..., n }, V = [n] = { 1,..., ṅ }, V = [n] { = 1,..., n }, V = V V = [n] [n] = [2n].,. E M(V ). (V, E). len(e) (V, E). V \ start(e) (V, E), V \ end(e)., Ě : Ě = { (v, u) K V \start(e),v \end(e) E u v. } K V. Remark 3.1. E M(V ), Ě : Ě M(V ), Ě E =, Ě E P(E), (V, E) (V, Ě E). 3

4 Remark 3.2. E M(V ), len(e) : len(e) = ((V, E) ) Ě., E P(V ), (i, j) E σ E (i) = j n S n σ E, E, len(e) σ E. (V, E) x = (x i,j ), weight monomial x E x E = (v,u) E. : Definition 3.3. K K V, det α (x, y; K) det α (x; K) : det α (x, y; K) = α n len(e) x E yě, det α (x; K) = E M(V,K) E P(V,K) x v,u α n len(e) x E., det α (x, y) = det α (x, y; K V ), det α (x) = det α (x; K V ). Remark 3.4. det α (x; K) = det α (x, 0; K). {,. { 1, 1},..., {ṅ, n} } K V, E V 0. E M (V ), Ě len(e ) : { } Ě = {v, u} K V \vertex(e ) E E 0 v u. len(e ) = ((V, E E 0) ) Ě. Remark 3.5. E M (V ), Ě : Ě M (V ), Ě E =, Ě E P (E ), (V, E E 0) (V, Ě E E 0). Remark 3.6. E M (V ), (V, E E 0)., (V, E E 0). E V \ vertex(e ) (V, E E 0). (V, E E 0) Ě. 4

5 (V, E ) x = (x i,j ), weight monomial x E x E = {v,u} E x v,u. v, u V x v,u = x u,v, x E well-defined. Definition 3.7. K K V, Hf α(x, y; K ) Hf α (x; K ) : Hf α (x, y; K ) = α n len(e ) x E, yě Hf α (x; K ) = E M (V,K ) E P (V,K ) α n len(e ) x E. Hf α (x, y) = Hf α (x, y; K V ), Hf α(x) = Hf α (x; K V ). Remark 3.8., Hf α (x; K ) = Hf α (x, 0; K ). Remark 3.9. A = (a ij ), α-determinant (or α-permanent) : det α (A) = σ S n α n len(σ) a 1,σ(1) a 2,σ(2) a n,σ(n). determinant permanent α-analogue ;, α-determinant α = 1 determinant, α = 1 permanent. (See also [13, 14].) Remark 3.2, α-determinant det α (A). [9] α-pfaffian, skew-symmetric matrix A : Pf α (A) = E P (V ) ( α) n len(e ) sgn(e )A E. sgn(e )A E E = { {x 1, x 1 },..., {x ṅ, x n } } x S 2n sgn(x)a x 1,x 1 a xṅ,x n, A skew symmetric sgn(e )A E x S 2n. α-pfaffian Pfaffian α-analogue,, α = 1 α-pfaffian Pfaffian Pf(A), i.e., sgn(x)a x 1 x 1 a xṅx n,., symmetric matrix B Hf α (B) Hf α (B) = α n len(e ) B E E P (V ),, α = 1, Hafnian Hf(B) = b x 1 x 1 b xṅx n. Hafnian analogue. 5

6 4 Main results det α (x, y), Hf α (x, y) Wishart. Propsition 4.1. W = (w i,j ) W p (ν, Σ, ),, W Wishart W p (ν, Σ, ). A B : a u,v = σ u,v, b u,v = δ u,v. : E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = ν n Hf ν 1(A, B) = Hf ν 1(νA, νb). : Theorem 4.2. A B : a u,v = σ iu,i v, b u,v = δ iu,i v. W W p (ν, Σ, ) : E[w i1,i 2 ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = ν n Hf ν 1(A, B) = Hf ν 1(νA, νb)., Wishart : Propsition 4.3. W = (w i,j ) CW p (ν, Σ, ), A, B : a u,v = σ u, v, b u,v = δ u, v. : E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = ν n det ν 1(A, B) = det ν 1(νA, νb)., : Theorem 4.4. A, B : a u,v = σ i u,i v, b u,v = δ i u,i v., W = (w i,j ) CW p (ν, Σ, ) : E[w i1,i 2 ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = ν n det ν 1(A, B) = det ν 1(νA, νb). Remark 4.5.,, Wishart,. [5, 12]. Remark 4.6. Wishart,, Wishart, Lu, Richards [7]; Graczyk, Letac, Massam [3, 4]; Vere-Jones [13]., Letac, Massam [6] Wishart,., Wishart,, Matsumoto [10]. 6

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