無限可積分系セッションアブストラクト
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1 7 Boel-Laplace I ) 1 ), ; xt + 1) λxt) bxt 1) = a[xt)] λ, b, a, t Boel-Laplace Boel-Laplace [ xt) = e Nζ1t N=1 N,N 1 N 1)! t N + l=1 l+n 1 ] l + N 1)! t l+n [1]) ζ 1 α ζ 1 = log α α = 1 N,N 1 b 1) 1,0 = α + bα 1 4a ) N,N 1 = a N 1 D N L=1 b N L) N L,N L 1 bl) L,L 1 N L 1 =0 N L 1 1) L + ) L+ L N D N = α N λ α N l+n 1 l+n+1 = l k=0 k+n,l+n 1 k+n,l+n 1 log α l+n 1 Kl+N α ± N log N log l + N)! K > 1 α < 1 α > 1 ; Reζ 1 t) 0, t R >> 1) :17K05371, 18K03418) 010 Mathematics Subject Classification: 44A10, 37D45 Henon map, Boel-Laplace tansfom, asymptotic expansion 1 hiaie.koichi.mu@ehime-u.ac.jp cmatsuoka@osaka-cu.ac.jp -43-
2 xt) xt) xt) f : C C P W s P ) W u P ) W s P ) = {Q C f n Q) P W u P ) = {Q C f n Q) P n )} n )} C C f- α α < 1 α > 1 x s t), x u t) W s P )\{P } = {x s t), bx s t 1)) t C}, W u P )\{P } = {x u t), bx u t 1)) t C}. f W s P ) \ {P } W u P ) \ {P } Λ f = W s P ) W u P ) f Λ f W s P ) W u P ) Q f = W s P ) W u P ) \ {P } Q f Λ f = Q f [1] C.Matsuoka an K.Hiaie, Special functions ceate by Boel-Laplace tansfom of Hénon map, Electon. Res. Announc. Math. Sci ), 1 11 [] C.Matsuoka an K.Hiaie, Entopy estimation of the Hénon attacto, Chaos, Solitons & Factals 45 01), [3] C.Matsuoka an K.Hiaie, Computation of entopy an Lyapunov exponent by a shift tansfom, Chaos 5, ). -44-
3 8 Boel-Laplace II ) 1 ) I xt) t C) xt) = L[X]t) = e ζt Xζ)ζ Laplace γ Xζ) xt) Boel γ AX = ax X + C, Aζ) = e ζ λ be ζ F G = ζ 0 F ζ ζ )Gζ )ζ convolution) C C = A0)X0) Xζ) Xζ) = a 0 + Xζ), a 0 : ) Xζ) A X + aa 0 X = W, W = W 0 a X X W 0 = aa 0ζ a 0 A + C) ζ path ζ ij i, j Z) Xζ) α α = 1) ζ 1 = log α Xζ) [1] Theoem 1. ζ = Nζ 1 +ξ N = 1,, ) Xζ) X N) ζ) X N) n = Xζ) = lim N m=0 m=0 X N) ζ), XN) ζ) = n=1 X n N) ζ), n,m+n 1ξ m+n 1 log ξ) n + eg n 1) ξ), 1 n N 1) n,m+n 1ξ m+n 1 log ξ) N + eg N 1) ξ), n N) :17K05371, 18K03418) 010 Mathematics Subject Classification: 44A10, 37D45 Henon map, Boel-Laplace tansfom, asymptotic expansion cmatsuoka@osaka-cu.ac.jp hiaie.koichi.mu@ehime-u.ac.jp -45-
4 W N) n 1 = m=0 m=0 v N) n 1,m+n 1ξ m+n 1 log ξ) n + eg n 1) ξ), n N) v N) n 1,m+n 1ξ m+n 1 log ξ) N + eg N 1) ξ). n N + 1) n 1 eg n 1) ξ) = R m ξ)log ξ) m, m=0 R m ξ) : n,m+n 1, v N) n 1,m+n 1 [1] 1 Xζ) Xζ) = a 0 + Xζ) Xζ) Laplace Theoem. xt) xt) = e ζt Xζ)ζ = γ 1 lim e ζt Xζ)ζ N γ N = lim N X R ζ, N) = πi) NN+1) πi) NN+1) 1 πi) NN+1) m=0 e iθ e iθ e ζt X R ζ, N)ζ, N,m+N 1 ξm+n 1. X R ζ, N) Xζ) θ Xζ) ζ Pat I xt) [1] Matsuoka C, Hiaie K 011 Special functions ceate by Boel-Laplace tansfom of Hénon map, Electo. Res. Ann. Math. Sci πi πi πi 4πi ζ 3 ζ 1 ζ ζ 3 ζ 11 ζ 1 ζ 13 0 ζ 01 ζ 0 ζ 03 ζ 04 ζ 11 ζ 1 ζ ζ
5 9 Constuction of two paametic efomation of KV-hieachy an solution by sigma function Victo Buchstabe ) 1 ) 1. X Q g X) = X g+1 + y 4 X g 1 y 6 X g + + y 4g X y 4g+ y 4, y 6,..., y 4g+ ) C g B g V g = {X, Y ) C Y = Q g X)} Sym V g ) V g [] Sym V g ) C 4 g = 3 KV V 3. Two paametic efomation of KV-hieachy y 4, y 6,..., y 4g+ ) C g FVg ) Vg Y 1 Q g X 1 ) Y Q g X ) C[X 1, Y 1, X, Y ] J g FV g ) C[X 1, Y 1, X, Y ]/J g FVg ) fx 1, Y 1, X, Y ) = fx, Y, X 1, Y 1 ) f CX 1, Y 1, X, Y ) FSym V g )) [] FSym V g )) FSym V g )) u = X 1 + X, u 4 = X 1 X ), u g 1 = Y 1 Y 4 L g) 1 g 3 = D D 1 ), X 1 X, u g+1 = Y 1 + Y, X 1 X L g) 1 g 1 = X D 1 X 1 D ), X 1 X D k = Y k Xk + Q gx k ) Yk, k = 1,. [], i = g 3, g 1 j =, 4, g 1, g+1l g) i u j u, u 4, u g 1, u g+1 Q[y 4, y 6,..., y 4g ] u, u 4, u g 1, u g+1 ) C 4 g = Dubovin g = 3 I) L 3) 3 u = u 5, L 3) 3 u 4 = u 7, L 3) 3 u 5 = 35u 4 4u u 4 3u 4 y 4 5u + u 4 ) + 4y 6 u y 8, L 3) 3 u 7 = 73u u 3 u 4 + 3u u 4) 10y 4 u 3 + u u 4 ) + y 6 3u + u 4 ) 3y 8 u + y 10, II) L 3) 5 u = u u 5 u 7, L 3) 5 u 4 = u u 7 u 4 u 5 ), L 3) 5 u 5 = u u 5 8u 3 u 4 18u u 4 8y 4 u u 4 + y 6 u + u 4 ) y 8 u + y 10, L 3) 5 u 7 = u 5 u 7 + 1u u 4 u 4 1u u 4 3u y 4 5u 4 u 4) 010 Mathematics Subject Classification: 14K5, 14H40, 14H4, 14H70 KV tayano7150@gmail.com 8 Gubkina St. Moscow, , Russia. -47-
6 y 6 3u 3 u u 4 ) + y 8 3u u 4 ) y 10 u. g = 3 T 1 = 1 L 3) 5, T 3 = L 3) 3 + X 1 + X L 3) 5 X 1 X X 1 X u = 4u, v = u 4 u ) w FSym V 3 )) w = T 1 w, ẇ = T 3 w 1 two paametic efome KV-hieachy v 4 u 4 u 6uu ) 3y 1 v u + 3y 14 vu 3u u) = 0, v 4 u 4u u u v) 3y 1 v v + 3y 14 vv 3u v) = 0, u = v, v = vu uv. y 1 = y 14 = 0 v 0 KV y 4,..., y 14 ) B 3 σw 1, w 3, w 5 ) V 3 σ i = wi σ σw 0) ) = 0, σ 1 w 0) ) 0, σ 5 w 0) ) 0 w 0) C 3 w 0) σw 1, w 3, φw 1, w 3 )) = 0 φ Ux, t) = σ 3x, t, φx, t)) σ 1 x, t, φx, t)), Kx, t) K = x K, V x, t) = σ 5x, t, φx, t)) σ 1 x, t, φx, t)) K = t K U, V two paametic efome KV-hieachy V 4 U 4 U 6UU ) 3y 1 V U + 3y 14 V U 3U U) = 0, V 4 U 4U U U V ) 3y 1 V V + 3y 14 V V 3U V ) = 0, U = V, V = V U UV. Remak 1 [3] sine-goon KV Remak V 3 y 1 = y 14 = 0 V 3 V, X, Y ) X, Y/X) FSym V 3 )) FSym V )) T 1, T 3 L ) 1, L ) 3 V L ) 1, L ) 3 V w1, w3 [1] T. Ayano, V. M. Buchstabe, Constuction of two paametic efomation of KVhieachy an solution in tems of meomophic functions on the sigma iviso of a hypeelliptic cuve of genus 3, axiv: , 018. [] V. M. Buchstabe, A.V. Mikhailov. Infinite imensional Lie algebas etemine by the space of symmetic squaes of hypeelliptic cuves, Functional Analysis an Its Applications, Volume 51, Issue 1, pp. 1, 017. [3] S. Matsutani, Relations of al functions ove subvaieties in a hypeelliptic Jacobian, CUBO A Math. J., Vol. 7, No 3, 005) 75 85, axiv:nlin/
7 10 KP ) [4, 5] jeu e taquin KP ectification) RSK [3] 1. jeu e taquin ectification) Young tableau) 1) : ) ) jeu e taquin 15 ) 1), inne cone) ), 3), 4), ) 3) : Jeu e taquin jeu e taquin ectification) Jeu e taquin [1] ) 010 Mathematics Subject Classification: 05E05, 37K40, KP, iwao@tokai.ac.jp -49-
8 . KP [4] [5] jeu e taquin KP KP ) [] ) R- jeu e taquin , 1, 0) 1, 0, 0) 1, 0, ) 1, 0, ) 1, 1, 3) []+ [] + [3] + [3] + [3] + [3], 0, 0) 1, 0, 0) 1, 1, 1) 1, 1, 1) 1,, ) = jeu e taquin ) RSK [3]) [1] W. Fulton. Young Tableaux: With Applications to Repesentation Theoy an Geomety. Lonon Mathematical Society Stuent Texts. Cambige Univesity Pess, [] S. Iwao. Topical integable systems an Young tableaux: shape equivalence an LittlewooRichason coesponence. Jounal of Integable Systems, 3 1), pp.xyy011, 018. [3] M. Noumi an Y. Yamaa. Topical Robinson-Schenste-Knuth coesponence an biational Weyl goup actions. In T. Shoji, M. Kashiwaa, N. Kawanaka, G. Lusztig, an K. Shinoa Es.), Repesentation theoy of algebaic goups an quantum goups, vol.40, pp Soc. Japan, Tokyo, 004. [4]. Skew Young jeu e taquin slie KP. Maste s thesis,, [5],. Jeu e taquin KP., 6AO-S, pp ,
9 11 Piei type fomulas fo the shifte Jack polynomials ) shifte Jack 1 shifte Jack ) shifte Jack Piei type fomulas., k Z >0,, α R, P patitions P := {m := m 1,..., m ) Z 0 m 1 m m }., m, n, k, x P, z C, e,k z) := z j1 z j, z) := k H J),k z) := S ) u; z) := 1 i 1< <i j l=0 l=0 ) k l I [] I =l H J),l z)u l, 1 i<j z i z j ), ) 1 z i zi z) z) i I J []\I I ) u; x) := J =k l ) z j zj, j J ) x k + ) u + k). k P, P k z; ) = Pk z1,..., z ; ) Jack. shifte Jack k z; ), 1) ip vanishing popety k m + δ; ) = 0 unless k m P k=1, ) ip k z; ) = P k z; ) + lowe tems) 1) ). Φ ) k z) := P ) ) ) k z; ), Ψ ) P k 1; k z) := P k 1; )Φ ) k k + k z) = P k z; k k [e a E0z) S ), E 0 z) := j=1 z j. δ; δ; u; z)]φ ) x 1 + z) = Φ ) x 1 + z)i ) u; x). ) Jack polynomials, shifte intepolation) Jack polynomials, Piei type fomulas g-shibukawa@math.kobe-u.ac.jp ) -51-
10 . 0 p, [ ] a E0 z)) p S ) u; z) p! Ψ ) k z) = J [] J =p, J c := [] \ J, ϵ j := δ ij ) 1 i Z, k J = k j J ϵ j h ) ±,J. x) := x j x k j k) ±, x j J,k J c j x k j k) z C, k P, k z + δ; ) ) I ) P k 1; u; z) =, k J = k + j J ϵ j. p=0 Ψ ) k J z)h ) +,J k J)I ) J c u; k). 3) ) I ) J u; k) := x c k + ) u + k) k J c J [] J =p k z + δ; ) J ) h ) P k J 1; +,J k)i) J u; c kj ). 4) 3. u j shifte Jack Piei type fomulas. 4. z C 0 j, ) ) k z; j ) k z; e P k 1;,j z) = J P k J 1; p=0 J [] J =p )h ) +,J k)e p,j p k J + δ ) J c ). 5) 5),, Jack 1), Jack Piei j, Φ ) k z)e,j z) = J [] J =j Φ ) z)h ) k J +,J k). 6) 3) u p. 6. z C 0 p, ) p [ ] a E0 z)) p H,p J) z) p! Ψ ) k z) = J [] J =p Ψ ) k J z)h ) +,J k J). [1] T. H. Koonwine : Okounkov s BC-type intepolation Maconal polynomials an thei q = 1 limit, Sém. Lotha. Combin, 7 014/15), 7pp. [] I. G. Maconal : Symmetic Functions an Hall Polynomials, Oxfo Univesity Pess,
11 1 Kostka polynomials with one column iagams of type B n, C n an D n ), ) 1. Fou-fol summation fomula fo Koonwine polynomials with one-column iagams n x = x 1,, x n ). a, b, c,, q, t Askey-Wilson/Koonwine. P 1 )x a, b, c, q, t) 1 ) 0 n) Koonwine. Definition 1.1. Lauent E x) : n yx i )1 y/x i ) = i=11 1) E x)y. 0 Theoem 1.. P 1 )x a, b, c, q, t) = 1) i+j E k l i j x)ĉ ek, l; t n +1+i+j )ĉ o i, j; t n +1 ), k,l,i,j 0 ĉ ek, l; s) = tc /a ; t ) k sc t; t ) k s c 4 /t ; t ) k 1/c ; t) l s/t; t) k+l 1 st k+l 1 a k c l, t ; t ) k sc /t; t ) k s a c /t; t ) k t; t) l sc ; t) k+l 1 st 1 ĉ o i, j; s) = a/b, s, sac/t, sa/t, sc/t, s a c/t 3 ; t) i t, s abc/t ; t) i s a c/t 3, s a c/t ; t ) i b i c/, ti s, t i sa /t, t i s a c /t 3 ; t) j t, t i s a c/t ; t) j t i s a c /t 3 ; t ) j j.. Kostka polynomials of type B n, C n an D n B n, B n ), C n, C n ), D n, D n ) Maconal, Koonwine : P Bn,Bn) λ x a; q, t) = P λ x q 1/, q 1/, 1, a q, t), P Cn,Cn) λ x b; q, t) = P λ x b 1/, b 1/, q 1/ b 1/, q 1/ b 1/ q, t), P Dn,Dn) λ x q, t) = P λ x 1, 1, q 1/, q 1/ q, t). P Bn,Bn) 1 ) x t; q, t), P Cn,Cn) 1 ) x t; q, t), P Dn,Dn) 1 ) x q, t) : P Bn,Bn) 1/t, t n +1, t n q, t n q/t; t) j 1 ) x t; q, t) = t) j t, t n +1 q; t) j t n q/t; t ) j j k=0 E x) = k t/q; t ) k t n ++j ; t ) k t n +j ; t ) k t ; t ) k t n +j ; t ) k t n +1+j q; t ) k t n +j ; t) k t n +1+j ; t) k 1 t n +k+j 1 t n +j q k E k j x), k=0 t n +1 ; t) k t n ; t) k q/t; t ) k t, t n + ; t ) k t n ; t ) k t n 1 q; t ) k t n 1 q; t ) k t k 1) j tn +k+1, t n +k q, t n +k q 1/, t n +k q 1/ ; t) j t n +4k q; t) j P Bn) 1 k j ) t; q, t), -53-
12 P Cn,Cn) 1/qt; t ) j t n, t ) j 1 t n +4j 1 ) x t; q, t) = qt) j E t ; t ) j qt n +3 ; t ) j 1 t n j x), E x) = qt; t ) j t n +j+, t ) j P Cn,Cn) t ; t ) j qt n +j+1 ; t 1 ) j ) j x t; q, t), P Dn,Dn) t/q; t ) j t n, t ) j 1 t n +j 1 ) x q, t) = q j E t ; t ) j qt n +1 ; t ) j 1 t n j x), E x) =, B n, C n, D n Schu s Bn) 1 ) q/t; t ) j t n +j+, t ) j 1 + t n t ; t ) j qt n +j 1 ; t ) j 1 + t n +j tj P Dn,Dn) x q, t). 1 j ) x), scn) 1 ) x), sdn) 1 ) s Bn) Bn,Bn) 1 ) x) = P 1 ) x q; q, q) = E x) + E 1 x), E x) = x) : 1) j s Bn) 1 j ) x), s Cn) Cn,Cn) 1 ) x) = P 1 ) x q; q, q) = E x) E x), E x) = s Cn) x), 1 j ) s Dn) Dn,Dn) 1 ) x) = P 1 ) x q, q) = E x). Definition.1. K Bn) t), 1 )1 j KCn) t), ) 1 ),1 j KDn) t) : ) 1 ),1 j ) s Bn) 1 ) x) = K Bn) 1 )1 j ) Bn,Bn) t)p1 j ) t; 0, t), scn) s Dn) 1 ) x) = K Dn) Dn,Dn) t)p x 0; t). 1 ),1 j ) 1 j ) 1 ) x) = K Cn) 1 ),1 j ) Cn,Cn) t)p1 j ) Theoem.. K Bn) t), 1 )1 j KCn) t), ) 1 )1 j KDn) t) : ) 1 )1 j ) K Bn) 1 )1 j ) t) = j k=0 K Cn) t) = [n + 1] t 1 )1 j ) tj [n + j + 1] t K Dn) t) = 1 + t n 1 )1 j ) tj 1 + t n +j ] = t n +j [ n + j 1 j 1 t n ; t ) k t n +1 ; t) j t k t, t n + ; t ) k t n ; t) k [ ] n + j j [ ] n + j j t [ ] n + j 1 + t j t j [n] q = 1 qn 1 q, [n] q! = [1] q [] q [n] q, t = j 1 k=0 t, [ ] m = j q x t; 0, t), t n + ; t ) k t n + ; t) j 1 t k, t, t n +4 ; t ) k t n +1 ; t) k [ ] [ ] n + j n + j j k=1 j t j 1 t, [m k + 1] q [m] q! = [k] q [j] q![m j] q!. -54-
λ n numbering Num(λ) Young numbering T i j T ij Young T (content) cont T (row word) word T µ n S n µ C(µ) 0.2. Young λ, µ n Kostka K µλ K µλ def = #{T
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More informationMacdonald, ,,, Macdonald. Macdonald,,,,,.,, Gauss,,.,, Lauricella A, B, C, D, Gelfand, A,., Heckman Opdam.,,,.,,., intersection,. Macdona
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More informationX G P G (X) G BG [X, BG] S 2 2 2 S 2 2 S 2 = { (x 1, x 2, x 3 ) R 3 x 2 1 + x 2 2 + x 2 3 = 1 } R 3 S 2 S 2 v x S 2 x x v(x) T x S 2 T x S 2 S 2 x T x S 2 = { ξ R 3 x ξ } R 3 T x S 2 S 2 x x T x S 2
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