Seeking for n! Derivatives

Transkript

1 Seeking for n! Derivatives $,000$ Reward

2 A remarkable determinant (,0) (,) (0,0) (0,) (,0) (,0) (,) (,) (0,0) (0,) (0,) General definition Δ μ (X, Y ) = det x p i j yq i n j i,j=

3 As a starter... n! dim M μ [X, Y] n! M μ [X, Y ]=L [ δ r x δ s y δ r x δ s y δ r n xn δ s n yn Δ μ (X, Y ) : r i,s i 0 ] For example ( using MAPLE) 6 independent derivatives! Could it be?????

4 The n! Conjecture ( ) dim M μ [X, Y ] = n! Proved by Mark Haiman using algebraic geometry,000$ Reward for an elementary proof Elementary means: By calculus or/and combinatorics give an algorithm that produces a triangular set of n! derivatives

5 The basic construction from a tableau to a point in n dimensional space α 6 α 7 α ( α α α α α α α β 5 β β β β β β ) x x β β β x x x 5 x 6 x 7 y y y y y 5 y 6 y 7 some polynomials vanishing at this orbit point x α x α x 7 α y β y 5 β Polynomials vanishing at all these tableau points: (x α )(x α )(x α ) (y 5 β )(y 5 β )(y 5 β ) (x α )(x α )(y β )

6 An Elementary Method (kicking) kicking for the shape x α x α y β (α,α,α ; β,β,β ) (α,α,α ; β,β,β ) y β (α,α,α ; β,β,β ) (α,α,α ; β,β,β ) y β (α,α,α ; β,β,β ) (α,α,α ; β,β,β ) y β x α y β x α (x α )(y β ) Δ [X, Y ] t t t t t + t+ t y Δ [X, Y ] x Δ [X, Y ] y Δ [X, Y ] x Δ [X, Y ] x y Δ [X, Y ] y + y AN INDEPENDENT SET WITH 6=! ELEMENTS

7 Kicking y β x α x α y β x α y β x α y β y β y β x α y β t t t t t t t t t t t (x α )(y β ) x α +6t +0t +6t + t The symmetry condition is satisfied y β y x α β y β t t t t t t t t t t t t

8 Kicking (x α ) (x α )(x α ) (y β ) (y β ) (y β ) (x α ) (x α ) (x α ) (x α ) (x α ) (x α ) t t t t t (x α ) +6t +0t +6t + t (x α ) Same polynomial q tq q (x α )(x α ) tq q tq (y β ) (y β ) (x α ) (y β ) t tq t tq t t q (y β ) (x α ) (y β ) t t q t t q t t q (y β )

9 Step The Algorithm Step Step Step 500 In this case Δ μ (X, Y ) = det x p i j yq i n j i,j= MAGICS? No Only Algebraic Combinatorics

10 Triangularity A typical term order The degree-lexicographic order and and Note: ldm(p ) < dlex ldm(p ) < dlex < dlex ldm(p m )

11 A basic tool Note: This proves the n! conjecture for λ = n (,0) (,0) (0,0)

12 Hilbert series its elements are called homogeneous of degree m R = H 0 (R) H (R) H (R) r s

13 Some Hilbert series Using Maple The dimension of 0 The dimension of 0

14 Flip

15 There is a formula for these Hilbert series! (brace yourself) F μ (q, t) = ν μ c μν (q, t)f ν (q, t) Some Basic Ingredients leg coarm coleg arm

16 Examples using MAPLE a simple recursive procedure How could such a messy recursion yield such beautiful polynomials? MAGICS? No Only Algebraic Combinatorics

17 A one way street with 8 parking spaces eight cars wish to park in it x f(x) The large number is their order of arrival The small number is their preferred parking place The corresponding Preference Function Each car proceeds to its preferred spot if occupied it parks in the available spot

18 A one way street with 8 parking spaces eight cars wish to park in it x f(x) The large number is their order of arrival The small number is their preferred parking place The corresponding Preference Function Each car proceeds to its preferred spot if occupied it parks in the available spot the number of cars that want to park in the first 5 places is less than 5... This preference function did not park the cars!!!

19 Parking functions and Dyck Paths A Parking function is a preference function that parks the cars

20 5 6 The sufficiency by induction

21 Two Statistics on Parking Functions area 5 dinv 6 t G n (q, t) = q PF PF n t area(pf) q dinv(pf) To see how beautiful we need MAPLE

22 The MAPLE data G n (q, t) = t area(pf) q dinv(pf) PF PF n t i G (q, t) = F (q, t) F (q, t) q j F (q, t) G 5 (q, t) = F 5 (q, t) Problem F μ (q, t) = PF PF μ t area(pf) q dinv(pf)

23 A simpler question? LI(a) denotes the length of the longest increasing subsequence Theorem #Π n,k = a C n,k sign(a) = k r=0 ( ) k ( ) k r n(n ) (n r +) r 00$ Reward for an elementary proof

24 Do you think that was enough? no here is more THE END

25 THE END