From multiline queues to Macdonald polynomials Sylvie Corteel (Paris - PowerPoint PPT Presentation

From multiline queues to Macdonald polynomials Sylvie Corteel (Paris Diderot), Olya Mandelshtam (Brown), and Lauren Williams (Harvard) olya@math.brown.edu FPSAC at Ljubljana, Slovenia July 4, 2019 1 0 3 Row 3 3 2 1 t 5 6 2 4 Row 2 2 2 3 2 6 1 2 7 8 3 3 6 1 2 7 8 3 4 5 Row 1 2 2 3 2 1 t 2 1 2

asymmetric simple exclusion process (ASEP) the ASEP is a particle process describing particles hopping on a finite 1D lattice: 1 particle per site, at each time step any two adjacent particles may swap with some probability, with possible interactions at the boundary t 1 α t β 1 1 t t 1 γ δ 1 2 2 t 3 2 1 t 0 0 0 multispecies ASEP on a ring: now we have particles of types 0 , 1 , . . . , L with J i particles of type i , represent the type by λ = ( L J L , . . . , 1 J 1 , 0 J 0 ). (Here λ = (3 , 2 , 2 , 2 , 1 , 0 , 0 , 0)) Markov chain with states that are rearrangements of the parts of λ , where possible transitions between states are swaps of adjacent particles: 1 X Y X Y A B B A

stationary probabilities 1 − 2 / 5 − t / 5 0 2 2 0 1 / 5 t / 5 1 1 · · · · · · 0 2 2 0 0 0 2 2 1 / 5 t / 5 1 1 t / 5 2 2 1 / 5 0 0 0 1 2 2 0 0 · · · · · · 2 1 / 5 t / 5 2 1 0 Pr(2 , 0 , 1 , 0 , 2) = 1 Pr(0 , 2 , 1 , 0 , 2) = 1 Z (3 + 7 t + 7 t 2 + 3 t 3 ) Z (5 + 6 t + 7 t 2 + 2 t 3 ) Pr(2 , 1 , 0 , 0 , 2) = 1 Pr(2 , 0 , 0 , 1 , 2) = 1 Z (6 + 7 t + 6 t 2 + t 3 ) Z (1 + 6 t + 7 t 2 + 6 t 3 ) Pr(2 , 1 , 2 , 0 , 0) = 1 Pr(2 , 0 , 1 , 2 , 0) = 1 Z (3 + 7 t + 7 t 2 + 3 t 3 ) Z (2 + 7 t + 6 t 2 + 5 t 3 ) (partition function) µ ˜ Z = � Pr ( µ )

ASEP and Macdonald polynomials symmetric Macdonald polynomial P λ ( x 1 , . . . , x n ; q , t ) defined by: � P λ = m λ + c µλ m µ , � P λ , P µ � = 0 if λ � = µ µ<λ Schur functions s λ at q = t Hall-Littlewood polynomials at q = 0 Jack polynomials at t = q α and q → 1 partition function of the ASEP on a ring at x 1 = · · · = x n = q = 1: � ˜ P λ (1 , . . . , 1; 1 , t ) = Pr ( µ ) µ (Cantini-de Gier-Wheeler ’15)

nonsymmetric Macdonald polynomials E µ ( x ; q , t ) E µ are simultaneous eigenfunctions of certain products of Demazure-Luztig operators, which are generators for the affine Hecke algebra of type A n − 1 : ( T i − t )( T i +1) = 0 , T i T i +1 T i = T i +1 T i T i +1 , T i T j = T j T i if | i − j | > 1 T i f = tf − tx i − x i +1 x i − x i +1 ( f − s i f ) Y i = T i · · · T n − 1 ω T − 1 · · · T − 1 i − 1 , Y i E µ = φ i ( µ ) E µ 1 E µ stabilize to P λ , specialize to Demazure characters at q = t = 0, specialize to key polynomials at q = t = ∞ . E µ (1 , . . . , 1; 1 , t ) = ˜ Pr ( µ ) when µ is a partition

probabilities of the ASEP with multiline queues Special case: t = 0 (Ferarri-Martin ’05) A multiline queue for particles of types 0 , 1 , . . . , L on an ASEP of n locations is a ball system on a cylinder of L rows and n columns Each ball picks the first available ball to pair with in the row below, weakly to its right The state of the multiline queue is read off Row 1 row 3 3 3 row 2 3 2 3 L row 1 3 2 1 1 3 3 0 2 1 1 3 µ = n Theorem (Ferrari-Martin ’05) Pr( µ )( t = 0) is proportional to the number of multiline queues with bottom row µ .

multiline queues for general t Combine a ball system with a queueing algorithm. Each ball chooses an available ball to pair with in the row below. t counts the number of available balls skipped: assign weight t total skipped (1 − t ). (1 − t ) The weight of each non-trivial pairing is t skipped 1 − t free . The state of the multiline queue is read off Row 1. j times: skipped = 3 j + 2 skipped row 3 3 3 t 2 (1 − t ) · (1 − t ) · t (1 − t ) j t 3 j +2 = t 2 (1 − t ) · 1 · 1 1 − t 3 1 − t 2 1 − t 4 (1 − t ) � (1 − t ) t 3 j +2 row 2 2 3 3 1 − t 3 t 3 (1 − t ) 4 = (1 − t 4 )(1 − t 3 )(1 − t 2 ) row 1 2 1 1 3 3 free µ = 2 1 0 1 3 3 t skipped (1 − t ) � wt( M ) = 1 − t free pairing Theorem (Martin ’18, Corteel-M-Williams ’18) Pr( µ ) = 1 � wt( M ) Z M ∈ MLQ( µ )

putting the “ q ” in the queue j x # balls in col j Define the x -weight of a queue M to be x M = � j Each pairing (of type ℓ , from row r ) that wraps around contributes q ℓ − r +1 Weight for each pairing is t skipped q ( ℓ − r +1) δ wrap 1 − t 1 − q ℓ − r +1 t free x M = x 2 1 x 2 2 x 3 x 2 4 x 5 x 2 row 3 3 3 3 6 qt 2 (1 − t ) · (1 − t ) 1 − qt 2 · 1 · t (1 − t ) j t 3 j +2 q j +1 = qt 2 (1 − t ) 1 − q 2 t 4 · 1 row 2 2 3 3 3 (1 − t ) � 1 − qt 3 1 − qt 3 qt 3 (1 − t ) 4 = row 1 2 1 1 3 3 (1 − q 2 t 4 )(1 − qt 3 )(1 − qt 2 ) µ = 2 1 0 1 3 3 1 − t q ( ℓ − r +1) δ wrap wt( M )( x ; q , t ) = x M t skipped � 1 − q ℓ − r +1 t free pairings Theorem (Corteel-M-Williams ’18) � E µ ( x ; q , t ) = wt( M )( x ; q , t ) when µ is a partition M ∈ MLQ( µ ) � P λ ( x ; q , t ) = wt( M )( x ; q , t ) M ∈ MLQ( λ )

proof We define f µ ( x ; q , t ) = � M ∈ MLQ( µ ) wt( M ) and show that: � f s i µ if µ i < µ i +1 T i f µ = tf µ if µ i = µ i +1 f µ 1 ,...,µ n ( x 1 , . . . , x n ) = q µ n f µ n ,µ 1 ,...,µ n − 1 ( qx n , x 1 , . . . , x n − 1 ) ( f µ and E µ are related by a triangular change of basis) thus: E µ = f µ when µ is a partition and � P λ = f µ µ

Koornwinder polynomials (Macdonald of type BC) α t 1 β γ δ Koornwinder polynomial K ( n − r , 0 ,..., 0) at q = t can be computed from the partition function Z n , r ( t ; α, β, γ, δ ) of the two-species ASEP with open boundaries (Corteel-Williams 2015, Cantini 2015) first combinatorial formula for certain special cases of Koornwinder polynomials using ASEP (Corteel-M-Williams 2016) Goal: compute nonsymmetric Kornwinder polynomials through multiline queues for the multispecies ASEP with open boundaries?