[PPT] - Weighted branching formulas for the hook lengths Matja Konvalinka PowerPoint Presentation

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Weighted branching formulas for the hook lengths

Matjaž Konvalinka Vanderbilt University (joint with Ionu¸ t Ciocan-Fontanine and Igor Pak)

FPSAC 2010, San Francisco, August 2010

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Partitions

Definition A partition λ of n is a finite sequence (λ1, λ2, . . . , λk) satisfying λ1 ≥ λ2 ≥ . . . ≥ λk > 0 and λ1 + . . . + λk = n. We represent a partition by its Young diagram [λ]. A corner of a partition is a square in the Young diagram that has no square below or to the right. The set

f all corners of λ is denoted C[λ].

Example

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 2 / 24

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Geometry of the Hilbert scheme of points

Okounkov Pandharipande relative DT theory of P1 × C2 relationship between QH∗

(C∗)2(Hilbn(C2)) and DT theory (CDKM)

Hilbert scheme is Geometric Invariant Theory quotient (via ADHM) machinery of abelian/nonabelian correspondence in Gromov-Witten theory an identity involving partitions equivariant quantum cohomology QH∗

(C∗)2(Hilbn)(C2) of the Hilbert scheme of points

DT = Donaldson and Thomas CDKM = Ciocan-Fontanine, Diaconescu, Kim and Maulik ADHM = Atiyah, Drinfeld, Hitchin and Manin

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Conjecture

Conjecture Choose variables α and β and a partition λ ⊢ n. For a square s = (i, j) of the Young diagram of λ, write ws = iα + jβ. Then we have

s∈[λ]

ws ·

t∈[λ]\{s,s+(1,1)}

(wt − ws − α)(wt − ws − β) (wt − ws − α − β)(wt − ws) = n(α + β).

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Equivalent formulation

It turns out that the conjecture is equivalent to the following. For positive integers x1, . . . , xℓ, y1, . . . , yℓ, the following is true:

ℓ

k=1

xk yℓ−k+1 k−1 p=1(xp+...+xk +yℓ−k+2+...+yℓ−p+1)· ℓ q=k+1(xk+1+...+xq+yℓ−q+1+...+yℓ−k+1) k−1 p=1(xp+1+...+xk +yℓ−k+2+...+yℓ−p+1)· ℓ q=k+1(xk+1+...+xq+yℓ−q+2+...+yℓ−k+1)

=

p+q≤ℓ+1

xpyq.

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 5 / 24

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Standard Young tableaux

Definition A Young tableau of shape λ ⊢ n is a bijective map map [λ] −→ [n]. A Young tableau is standard if the tableau is increasing in rows and in columns. The number of standard Young tableaux of shape λ is denoted by f λ. Example We have f 32 = 5:

3 1 3 5 2 4 1 3 4 2 5 1 2 5 3 4 1 2 5 4 3 1 2 4 5

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 6 / 24

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Hook-length formula

Definition For a partition λ and a square (i, j) of [λ], define the hook as the squares weakly to the right of or below (i, j). The hook length hij is the number of squares in Hij. Example For λ = 32, the hook lengths are 4, 3, 2, 1, 1:

1 4 3 1 2

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Hook-length formula

Theorem The number of standard Young tableau of shape λ is f λ = n!

hij

. Example The number of standard Young tableau of shape 32 is f 32 = 5! 4 · 3 · 2 · 1 · 1 = 5.

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Two examples

2 5 4 6 7 1 3 1 2 7 6 5 3 4 8

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Two examples

1 2 7 6 5 3 4 8 2 5 4 6 7 3 1

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 10 / 24

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Two examples

1 2 7 6 5 3 4 8 2 5 4 6 7 3 1 n k

=

(n + 1)! (n + 1)k!(n − k)! Cn = (2n)! n!(n + 1)!

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 10 / 24

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Greene-Nijenhuis-Wilf proof

In a SYT of shape λ, n must be in one of the corners, which implies f λ =

c∈C[λ]

f λ−c. By induction, it suffices to show that F λ =

c∈C[λ]

F λ−c, where F λ = n!

hij
r, equivalently, that
c∈C[λ]

F λ−c F λ = 1.

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 11 / 24

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Branching formula

This last formula is equivalent to

(r,s)∈C[λ]

1 n

r−1

i=1
1 +

1 his − 1 s−1

j=1
1 +

1 hrj − 1

= 1
r

n ·

(i,j)∈[λ]\C[λ]

(hij − 1) =

(r,s)∈C[λ]

   

(i,j)∈[λ]\C[λ]

i=r,j=s

(hij − 1)    

r−1

i=1

his

s−1

j=1

hrj.

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 12 / 24

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Branching formula

This last formula is equivalent to

(r,s)∈C[λ]

1 n

r−1

i=1
1 +

1 his − 1 s−1

j=1
1 +

1 hrj − 1

= 1
r

n ·

(i,j)∈[λ]\C[λ]

(hij − 1) =

(r,s)∈C[λ]

   

(i,j)∈[λ]\C[λ]

i=r,j=s

(hij − 1)    

r−1

i=1

his

s−1

j=1

hrj. This is the branching rule for the hook lengths. The former version can be proved by constructing a random process with terms in the sum on the left-hand side denoting probabilities of all possible

utcomes (the hook walk).

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 12 / 24

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Observation

When λ = (ℓ, ℓ − 1, . . . , 1), this is our identity when all xi, yj are 1. So we are trying to prove a weighted version of the branching rule for the hook lengths for the staircase shape.

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 13 / 24

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Weighted hooks

n ·

(i,j)∈[λ]\C[λ]

(hij − 1) =

(r,s)∈C[λ]

   

(i,j)∈[λ]\C[λ]

i=r,j=s

(hij − 1)    

r−1

i=1

his

s−1

j=1

hrj.

x1 x2 x3 x4 x4 x5 x5 x6 x6 y1 y2 y3 y3 y4 y4 y5 y5 y6 y6 y7 y7

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 14 / 24

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Weighted branching rule for hook lengths

Theorem  

(p,q)∈[λ]

xpyq   ·   

(i,j)∈[λ]\C[λ]
xi+1 + . . . + xλ′

j + yj+1 + . . . + yλi



  =

(r,s)∈C[λ]

xrys    

(i,j)∈[λ]\C[λ]

i=r,j=s

xi+1 + . . . + xλ′

j + yj+1 + . . . + yλi



   ×  

r−1

i=1

(xi + . . . + xr + ys+1 + . . . + yλi)   ×  

s−1

j=1
yj + . . . + ys + xr+1 + . . . + xλ′

j

 

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 15 / 24

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Example

x1 x2 x3 x4 y1 y2 y3

(x1y1 + x2y1 + x3y1 + x4y1 + x1y2 + x2y2 + x1y3) (x2 + x3 + x4 + y2 + y3)(x2 + y3)(x3 + x4 + y2)x4 = x1y3

x2 + x3 + x4 + y1 + y2 + y3
(x3 + x4 + y2)
x2 + y2 + y3
x4+

x2y2 (x2 + x3 + x4 + y2 + y3)

x3 + x4 + y1 + y2

x1 + x2 + y3

x4+

x4y1

x1 + x2 + x3 + x4 + y2 + y3

x2 + x3 + x4 + y2

(x2 + y3)
x3 + x4
Matjaž Konvalinka (Vanderbilt University)

Weighted branching formulas August 2010 16 / 24

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Interpretation of the left-hand side

 

(p,q)∈[λ]

xpyq   ·   

(i,j)∈[λ]\C[λ]
xi+1 + . . . + xλ′

j + yj+1 + . . . + yλi



  ◮ special labels xp, yq ◮ a label xk for some i < k ≤ λ′

j , or yl for some j < l ≤ λi, in every

non-corner square (i, j)

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 17 / 24

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Example

x2 x3 x3 x4 x4 x4 x4 x4 x4 x5 x5 x5 x5 x6 x6 y1 y2 y3 y3 y3 y3 y4 y4 y5 y5 y5 y6 y6 y6 y6 y6 y6 y6 y7 y7 y7

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 18 / 24

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Interpretation of the right-hand side

(r,s)∈C[λ]

xrys

(i,j)∈[λ]\C[λ]

i=r,j=s

xi+1 + . . . + xλ′

j + yj+1 + . . . + yλi

r−1
i=1

(xi + . . . + xr + ys+1 + . . . + yλi)

s−1

j=1
yj + . . . + ys + xr+1 + . . . + xλ′

j

◮ special labels xr, ys, corresponding to the corner (r, s)

◮ a label xk for some i < k ≤ λ′

j , or yl for some j < l ≤ λi, in every

non-corner square (i, j), i = r, j = s ◮ a label xk for some i ≤ k ≤ λ′

j , or yl for some s < l ≤ λi, in every

non-corner square (i, s) ◮ a label xk for some r < k ≤ λ′

j , or yl for some j ≤ l ≤ λi, in every

non-corner square (r, j)

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 19 / 24

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Example

x2 x3 x3 x4 x4 x4 x4 x4 x4 x5 x5 x5 x5 x6 x6 y1 y2 y3 y3 y3 y3 y4 y4 y5 y5 y5 y6 y6 y6 y6 y6 y6 y6 y7 y7 y7

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 20 / 24

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Hook walk

x2 x3 x3 x4 x4 x4 x4 x4 x4 x5 x5 x5 x5 x6 x6 y1 y2 y3 y3 y3 y3 y4 y4 y5 y5 y5 y6 y6 y6 y6 y6 y6 y6 y7 y7 y7

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Shifting the labels

x2 x3 x3 x4 x4 x4 x4 x4 x4 x5 x5 x5 x5 x6 x6 y1 y2 y3 y3 y3 y3 y4 y4 y5 y5 y5 y6 y6 y6 y6 y6 y6 y6 y7 y7 y7

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Example

x2 x2 x3 x3 x3 x3 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x4 x5 x5 x5 x5 x5 x5 x5 x5 x6 x6 x6 x6 y1 y1 y2 y2 y3 y3 y3 y3 y3 y3 y3 y3 y4 y4 y4 y4 y5 y5 y5 y5 y5 y5 y6 y6 y6 y6 y6 y6 y6 y6 y6 y6 y6 y6 y6 y6 y7 y7 y7 y7 y7 y7

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 23 / 24

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Final remarks

◮ weighted hook walk ◮ application: Kerov’s q-walk ◮ application: generation of random SYT with probabilities certain rational functions in xi, yj ◮ variants ◮ complementary formulas ◮ shifted diagrams ◮ d-complete posets?

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 24 / 24

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Final remarks

◮ weighted hook walk ◮ application: Kerov’s q-walk (but not Garsia-Haiman’s q, t-walk) ◮ application: generation of random SYT with probabilities certain rational functions in xi, yj ◮ variants ◮ complementary formulas ◮ shifted diagrams ◮ d-complete posets?

Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 24 / 24