Tomoyuki Shirai Institute of Mathematics for Industry, Kyushu University 1 Schur Schur Okounkov [14] Okounkov-Reshetikhin [16] Young Young
Schur Macdonald [2] Macdonald integrable probability [4] Schur A. Borodin Borodin-Gorin 2
Plancherel
Yn n
Plancherel P(n)
P Pl
Sn
RIMS . (

3
Schur
N ~ Po(t)
{hk}k=1 Schur ( x 1 , x 2 , . . . ) 3.1. sι(ρ) ≥ 0 (Αγ ∈ γ) 3.2 (Schur H(ρι;ρ2) 3.3. (i) H(ρι;ρ2) < ∞ (ii) H(ρι;ρ2) = ∞ 3.4. Schur ρ 0), β = (β1 ≥ β2 ≥ ···≥ 0), γ≥ 0 α = (α1 ≥ α2 ≥ ··· ≥ ¿ Schur(      ) hk
Thoma(1964)
S ∞
total positivity totally positive A Schoenberg
{ an } n=0,1,...
totally positive [8]
3.6. c ≥ 0 (i) α1 = c 0 ρ(c)
0 P(c) (ii) βι = c
X λ (iii) γ = c
pe (iii)
Schur Ppp1 = Spo;po Schur Okounkov 3.7 ([14]). Schur (determinantal point process)
K : R × R → C ρ({x,y}) x,y ∈ R
0

4
Schur
(Schur process)
S'
μχ(x 1, . . ., x n ) := P(X t 1 = x 1, X t 2 = x 2, . . . , X t n = x n )   (∀ n ≥ 1, ∀ t 1, . . ., t n ∈ T,∀ x 1, . . .,x n ∈ S )
S λ / μ

5
4.1 (Schur [16]). 2N Schur ρ † , i = 0, 1, . . . , N − 1, ρ − , j = 1, 2, . . . , N
ß ⊂ λ(¹) ⊃ μ(¹) ⊂ λ S Schur

1: Young
4.2. (i) N = 1 Schur Schur Sp+;p- (ii)  ક્ λ/μ (0) = δλ,μ Young (iii) ) (N) Sベ0 0 0 2: Z 4.1
3
Young
Youngも
 
 45 3 Young 1
3 Young q ∈ (0,1) n i, j Vn = {A = (aij)1<i,j<n : aij ∈ Z≥0(Vi,j),aij }
χ(0) = χ(2n) = ∅
Voo
MacMahon [13] 4.2
Gelfand-Tsetlin pattern
Schur 1 ρ i = 0 (i = 1,2,...,N) 2
S

9
lacing inter-
Schur
5
Schur (cf. [1]) 5.1 S'
{Xt}t=0,1,... (5.1) P(X t+1 = y | X t = x, X t −1 = x t −1, . . . , X 0 = x 0) = P(X t+1 = y | X t = x )(=: P(x,y)) P = (P(x,y))x,yES P S μ
μ P n
元P t (x,y) Si, Ši (i = 1, 2, . . . , n ) , Λ k − 1 : S k × S k − 1 → [0, 1], ˜ ∑ k − 1 : Š k × Š k − 1 → [0, 1] (k = 2, 3, . . . , n ), P k : S k × Š k → [0, 1] (k = 1, 2, . . . , n )
)

5.2
Schur
Schur
4. (ρ,ρ',ρ'') s λ ((ρ,ρ')) рк(ρ,ρ′), ςς(ρ,ρ′) (5.2) X2 3. ρ, ρ′ (5.5) (5.6) 5.3 Y Schur Y Y
· Σμ ρ λμ(ρ;ρ') = Σμ ρ λμ(ρ;ρ') = 1
V
(resp.
5.4. μ E Y
Schur
Sp1;p2;p3P1(p2;p3) = Sp1;p2;
( Schur 5.5. 4.2 Schur
S(X(1), X(2), . . . , X(N-1), X(N)) ∑ 'x(1)(P t ) 'x(2)/x(1)(P t ) ··· 'x(N -1)(P t -1) 'x(N)(P n ) (5.7) = Sρ$.ρ$,….ρ$_1;ρ==(λ(N))ρ$(N),χ(N-1)(ρ$,ρ$,ρ$,ρ$_2;ρ$_1) · · · ρ$(2),χ(1)(ρ$,ρ$)
5.4 YN (5.10) Λ k_1 Pk = Pk_1 Λ k_1 · Schur 5.6.   P1, P2, P3 P↑(ρι,ρ2;ρ3)P↓(ρ1;ρ2) = P↓(ρ1;ρ2)P↑(ρ1;ρ3). . (5.6) (5.11) 5.1 N (5.9) ● WN

Gelfand-Tsetlin pattern GT N
5.5
(5.2) P χ(k),μ(k) (P † , · · · , P † − 1; P ′ )P † (k),μ(k− 1) (P † , P † , · · · , P † − 2; P † − 1) = P t (k),µ(k) (kρ(1) ; ρ(b) )P t (k-1) ((k-1) ((k-1)ρ(1) ; ρ(1) ). X(k) ⊂ µ(k) vertical strip P χ(k),μ(k) (kρ(1); ρ(b) > 0 μ(k)つμ(k-1) 1)ρ(¹); ρ(¹) > 0 horizontal strip 4.3 X(k)(t) k ( λ ( 1 )( t ), λ ( 2 )( t ), . . . , λ ( N )( t )) ∈ Ψ N
TASEP(totally asymmetric simple x(N)(t) < x(N-1)(t) < ··· < x(1)(t) < x(1)(t)
15
6: GT10 t = 0 t = 8 .
Charlier
≤ x ( N −1) ( t ) ≤
  PushTASEP 1 5.9. N Dyson (GUE ) TASEP Schur Gelfand-Tsetlin pattern 6 integrable probability Schur stochastic six-vertex model Macdonald [3] KPZ(Kardar-Parisi-Zhang) KPZ [5, 6, 9] Schur KPZ [1] A. Borodin, Schur dynamics of the Schur processes, Adv. Math. 228 (2011), 2268–2291. [2] A. Borodin and I. Corwin, Macdonald processes, available at http://arxiv.org/abs/ 1111.4408 [3] A. Borodin, I. Corwin and V. Gorin, Stochastic six-vertex model, available at http:// arxiv.org/abs/1407.6729 [4] A. Borodin and V. Gorin, Lectures on integral probability, available at http://arxiv.org/ abs/1212.3351 [5] I. Corwin, The Kardar-Parisi-Zhang equation and universality class, Rand. Mat.: Theo. Appl. 1:1130001(2012). [6] I. Corwin, Kardar-Parisi-Zhang universality class, Notice of the AMS, March 2016, 230–239. [7] A. Edrei, On the generating function of a doublyinfinite, totally positive sequence, Trans. Amer. Math. Soc. 74 (1953), no. 3, 367–383. [8] A. Edrei, Proof of a conjecture of Schoenberg on the generating function of a doublyinfinite, totally positive sequence, Canadian J. Math. 5 (1953), 86–94. [9] K. Kardar, G. Parisi and Y. Z. Zhang, Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56 (1986), 889–892. [10] S. Karlin, Total positivity. Vol. I. Stanford University Press, Stanford, Calif 1968 xii+576 pp. [11] S. V. Kerov, Asymptotic representation theory of the symmetric group and its applica- tions in analysis, Translations of Mathematical Monographs, 219. American Mathematical Society, 2003. [12] I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, 1995. [13]  P. .A. MacMahon, Combinatory Analysis. Cambridge University Press, 1915-1916. reprinted by Chelsea Publishing Company, New York, 1960. [14] A. Okounkov, Infinite wedge and random partitions, Selecta Math. 7 (2001), 57–81. [15] A. Okounkov, Symmetric functions and random partitions, available at http://arxiv. org/abs/math/0309074v1 [16] A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), 581–603. [17] D. Romik, The Surprising Mathematics of Longest Increasing Subsequences, Cambridge University Press, 2015. [18] B. E. Sagan, The symmetric group. Representations, combinatorial algorithms, and sym- metric functions. Second edition. Graduate Texts in Mathematics 203 , Springer-Verlag, 2001. [19] I. J. Schoenberg, Some analytical aspects of the problem of smoothing, in: Courant An- niversary Volume"  Studies and Essays " , New York, 1948, pp. 351-370. [20] T. Shirai and Y. Takahashi, Random point fields associted with certain Fredholm determi- nants (I): fermion, Poisson and boson point processes, J. Funct. Anal. 205 (2003), 414–463. [21] T. Shirai and Y. Takahashi, Random point fields associted with certain Fredholm determi- nants (II): fermion shifts and their ergodic properties (with Y.Takahashi), Annals of Prob. 31 (2003), 1533–1564. [22] A. Soshnikov, Determinantal random point fields, Russian Math. Surveys, 55 (2000), 923– 975. [23] E. Thoma, Die unzerlegbaren, positive-definiten Klassenfuntionen der abzählbar un- endlichen, symmetrischen Gruppe, Math. Zeitschr. 85 (1964), 40–61. [24] 2015