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

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

Partitions Definition A partition λ of n is a finite sequence ( λ 1 , λ 2 , . . . , λ k ) satisfying λ 1 ≥ λ 2 ≥ . . . ≥ λ k > 0 and λ 1 + . . . + λ k = 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 of all corners of λ is denoted C [ λ ] . Example Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 2 / 24

Geometry of the Hilbert scheme of points ( C ∗ ) 2 (Hilb n )( C 2 ) of the Hilbert scheme of points equivariant quantum cohomology QH ∗ an identity involving partitions machinery of abelian/nonabelian correspondence in Gromov-Witten theory Okounkov Hilbert scheme is Geometric Invariant Theory quotient (via ADHM) Pandharipande ( C ∗ ) 2 (Hilb n ( C 2 )) and DT theory (CDKM) relationship between QH ∗ relative DT theory of P 1 × C 2 DT = Donaldson and Thomas CDKM = Ciocan-Fontanine, Diaconescu, Kim and Maulik ADHM = Atiyah, Drinfeld, Hitchin and Manin Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 3 / 24

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 w s = i α + j β . Then we have ( w t − w s − α )( w t − w s − β ) � � w s · = n ( α + β ) . ( w t − w s − α − β )( w t − w s ) s ∈ [ λ ] t ∈ [ λ ] \ { s , s +( 1 , 1 ) } Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 4 / 24

Equivalent formulation It turns out that the conjecture is equivalent to the following. For positive integers x 1 , . . . , x ℓ , y 1 , . . . , y ℓ , the following is true: ℓ � k − 1 p = 1 ( xp + ... + xk + y ℓ − k + 2 + ... + y ℓ − p + 1 ) · � ℓ q = k + 1 ( xk + 1 + ... + xq + y ℓ − q + 1 + ... + y ℓ − k + 1 ) � xk y ℓ − k + 1 � k − 1 p = 1 ( xp + 1 + ... + xk + y ℓ − k + 2 + ... + y ℓ − p + 1 ) · � ℓ q = k + 1 ( xk + 1 + ... + xq + y ℓ − q + 2 + ... + y ℓ − k + 1 ) k = 1 � = x p y q . p + q ≤ ℓ + 1 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 5 / 24

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: 1 2 3 1 2 4 1 2 5 4 5 3 5 3 4 1 3 4 1 3 5 2 5 2 4 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 6 / 24

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 h ij is the number of squares in H ij . Example For λ = 32, the hook lengths are 4 , 3 , 2 , 1 , 1: 4 3 1 2 1 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 7 / 24

Hook-length formula Theorem The number of standard Young tableau of shape λ is n ! f λ = . � h ij Example The number of standard Young tableau of shape 32 is 5 ! f 32 = = 5 . 4 · 3 · 2 · 1 · 1 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 8 / 24

Two examples 0 1 4 6 7 1 2 4 7 2 3 5 6 8 3 5 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 9 / 24

Two examples 0 1 4 6 7 1 2 4 7 2 3 5 6 8 3 5 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 10 / 24

Two examples 0 1 4 6 7 1 2 4 7 2 3 5 6 8 3 5 � n � ( n + 1 )! ( 2 n )! = C n = k ( n + 1 ) k !( n − k )! n !( n + 1 )! Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 10 / 24

Greene-Nijenhuis-Wilf proof In a SYT of shape λ , n must be in one of the corners, which implies f λ = � f λ − c . c ∈ C [ λ ] By induction, it suffices to show that n ! F λ = � F λ = F λ − c , where � h ij c ∈ C [ λ ] or, equivalently, that F λ − c � = 1 . F λ c ∈ C [ λ ] Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 11 / 24

Branching formula This last formula is equivalent to � s − 1 r − 1 � � 1 1 1 � � � � 1 + 1 + = 1 n h is − 1 h rj − 1 ( r , s ) ∈ C [ λ ] i = 1 j = 1 or   r − 1 s − 1 � � � � �   n · ( h ij − 1 ) = ( h ij − 1 ) h is h rj .     ( i , j ) ∈ [ λ ] \ C [ λ ] ( r , s ) ∈ C [ λ ] i = 1 j = 1 ( i , j ) ∈ [ λ ] \ C [ λ ] i � = r , j � = s Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 12 / 24

Branching formula This last formula is equivalent to � s − 1 r − 1 � � 1 1 1 � � � � 1 + 1 + = 1 n h is − 1 h rj − 1 ( r , s ) ∈ C [ λ ] i = 1 j = 1 or   r − 1 s − 1 � � � � �   n · ( h ij − 1 ) = ( h ij − 1 ) h is h rj .     ( i , j ) ∈ [ λ ] \ C [ λ ] ( r , s ) ∈ C [ λ ] i = 1 j = 1 ( i , j ) ∈ [ λ ] \ C [ λ ] i � = r , j � = s 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 outcomes (the hook walk). Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 12 / 24

Observation When λ = ( ℓ, ℓ − 1 , . . . , 1 ) , this is our identity when all x i , y j 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

Weighted hooks   r − 1 s − 1 � � � � �   n · ( h ij − 1 ) = ( h ij − 1 ) h is h rj .     ( i , j ) ∈ [ λ ] \ C [ λ ] ( r , s ) ∈ C [ λ ] i = 1 j = 1 ( i , j ) ∈ [ λ ] \ C [ λ ] i � = r , j � = s y 1 y 2 y 3 y 4 y 5 y 6 y 7 x 1 x 2 y 7 x 3 y 3 y 4 y 5 y 6 x 4 x 4 x 5 x 5 x 6 x 6 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 14 / 24

Weighted branching rule for hook lengths Theorem     � � � �  · x p y q x i + 1 + . . . + x λ ′ j + y j + 1 + . . . + y λ i      ( p , q ) ∈ [ λ ] ( i , j ) ∈ [ λ ] \ C [ λ ]   � � � �   = x r y s x i + 1 + . . . + x λ ′ j + y j + 1 + . . . + y λ i     ( r , s ) ∈ C [ λ ] ( i , j ) ∈ [ λ ] \ C [ λ ] i � = r , j � = s   r − 1 � ( x i + . . . + x r + y s + 1 + . . . + y λ i ) ×   i = 1  � s − 1 � � × y j + . . . + y s + x r + 1 + . . . + x λ ′   j j = 1 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 15 / 24

Example y 1 y 2 y 3 x 1 x 2 x 3 x 4 ( x 1 y 1 + x 2 y 1 + x 3 y 1 + x 4 y 1 + x 1 y 2 + x 2 y 2 + x 1 y 3 ) ( x 2 + x 3 + x 4 + y 2 + y 3 )( x 2 + y 3 )( x 3 + x 4 + y 2 ) x 4 = � � � � ( x 3 + x 4 + y 2 ) x 1 y 3 x 2 + x 3 + x 4 + y 1 + y 2 + y 3 x 2 + y 2 + y 3 x 4 + � � � � x 2 y 2 ( x 2 + x 3 + x 4 + y 2 + y 3 ) x 3 + x 4 + y 1 + y 2 x 1 + x 2 + y 3 x 4 + � � � � � � ( x 2 + y 3 ) x 4 y 1 x 1 + x 2 + x 3 + x 4 + y 2 + y 3 x 2 + x 3 + x 4 + y 2 x 3 + x 4 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 16 / 24

Interpretation of the left-hand side     � � � �  · x p y q x i + 1 + . . . + x λ ′ j + y j + 1 + . . . + y λ i      ( p , q ) ∈ [ λ ] ( i , j ) ∈ [ λ ] \ C [ λ ] ◮ special labels x p , y q ◮ a label x k for some i < k ≤ λ ′ j , or y l for some j < l ≤ λ i , in every non-corner square ( i , j ) Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 17 / 24

Example y 1 x 3 x 6 y 4 y 6 y 6 x 4 x 4 x 2 y 3 x 4 y 4 x 3 y 7 y 7 x 4 y 6 x 5 y 5 y 5 x 5 x 4 x 4 y 6 y 3 x 5 y 6 x 5 y 7 y 6 y 3 x 6 y 5 y 6 y 2 y 3 Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 18 / 24

Interpretation of the right-hand side � � � � x r y s x i + 1 + . . . + x λ ′ j + y j + 1 + . . . + y λ i ( r , s ) ∈ C [ λ ] ( i , j ) ∈ [ λ ] \ C [ λ ] i � = r , j � = s r − 1 s − 1 � � � � ( x i + . . . + x r + y s + 1 + . . . + y λ i ) y j + . . . + y s + x r + 1 + . . . + x λ ′ j i = 1 j = 1 ◮ special labels x r , y s , corresponding to the corner ( r , s ) ◮ a label x k for some i < k ≤ λ ′ j , or y l for some j < l ≤ λ i , in every non-corner square ( i , j ) , i � = r , j � = s ◮ a label x k for some i ≤ k ≤ λ ′ j , or y l for some s < l ≤ λ i , in every non-corner square ( i , s ) ◮ a label x k for some r < k ≤ λ ′ j , or y l for some j ≤ l ≤ λ i , in every non-corner square ( r , j ) Matjaž Konvalinka (Vanderbilt University) Weighted branching formulas August 2010 19 / 24