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Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Daping Weng Yale University March 2018 Joint work with Jiuzu Hong and Linhui Shen

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Table of Contents 1 Cyclic Sieving Phenomenon of Plane Partitions 2 Decorated Grassmannian and Decorated Configuration Space 3 Cluster Duality of Grassmannian 4 Proof of CSP of Plane Partitions

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Table of Contents 1 Cyclic Sieving Phenomenon of Plane Partitions 2 Decorated Grassmannian and Decorated Configuration Space 3 Cluster Duality of Grassmannian 4 Proof of CSP of Plane Partitions Throughout this talk, let a , b , c be three positive integers and let n := a + b .

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Cyclic Sieving Phenomenon Definition Let S be a finite set. Let g be a permutation on S that is of order m . Let F ( q ) be a polynomial in q . We say that the triple ( S , g , F ( q )) exhibits the cyclic sieving phenomenon (CSP) if the fixed point set cardinality # S g d is equal to the polynomial evaluation F ( ζ d ) for all d ≥ 0 where ζ is a primitive m th root of unity.

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Cyclic Sieving Phenomenon Definition Let S be a finite set. Let g be a permutation on S that is of order m . Let F ( q ) be a polynomial in q . We say that the triple ( S , g , F ( q )) exhibits the cyclic sieving phenomenon (CSP) if the fixed point set cardinality # S g d is equal to the polynomial evaluation F ( ζ d ) for all d ≥ 0 where ζ is a primitive m th root of unity. Example � [ n ] Let [ n ] := { 1 , . . . , n } and let � be the set of k -element subsets of [ n ]. k Consider the cyclic shift R : i �→ i + 1 mod n on [ n ] and the induced action on �� [ n ] � � [ n ] � n � n � � � � . It is known that the triple , R , exhibits CSP, where q is the k k k k q quantum binomial coefficient.

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Cyclic Sieving Phenomenon Definition Let S be a finite set. Let g be a permutation on S that is of order m . Let F ( q ) be a polynomial in q . We say that the triple ( S , g , F ( q )) exhibits the cyclic sieving phenomenon (CSP) if the fixed point set cardinality # S g d is equal to the polynomial evaluation F ( ζ d ) for all d ≥ 0 where ζ is a primitive m th root of unity. Example � [ n ] Let [ n ] := { 1 , . . . , n } and let � be the set of k -element subsets of [ n ]. k Consider the cyclic shift R : i �→ i + 1 mod n on [ n ] and the induced action on �� [ n ] � � [ n ] � n � n � � � � . It is known that the triple , R , exhibits CSP, where q is the k k k k q quantum binomial coefficient. Although the definition of CSP seems very combinatorial, many proofs of known CSP involve quite a bit of geometric representation theory. Please see Sagan’s survey [Sag11] for more detailed examples.

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Definition An a × b plane partition is an a × b matrix π with non-negative integer entries such that every row is non-increasing from left to right and every column is non-increasing from top to bottom.

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Definition An a × b plane partition is an a × b matrix π with non-negative integer entries such that every row is non-increasing from left to right and every column is non-increasing from top to bottom. Example Here’s an example of a 2 × 3 plane partition. 3 2 2 3 1 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Definition An a × b plane partition is an a × b matrix π with non-negative integer entries such that every row is non-increasing from left to right and every column is non-increasing from top to bottom. Example Here’s an example of a 2 × 3 plane partition. 3 2 2 3 1 0 Remark. Think of a plane partition as a 3d Young diagram.

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Denote the collection of a × b plane partitions with entries no bigger than some c > 0 by P ( a , b , c ).

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Denote the collection of a × b plane partitions with entries no bigger than some c > 0 by P ( a , b , c ). For a plane partition π , define � | π | := π i , j . i , j

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Denote the collection of a × b plane partitions with entries no bigger than some c > 0 by P ( a , b , c ). For a plane partition π , define � | π | := π i , j . i , j For any triple ( a , b , c ), define � q | π | . M a , b , c ( q ) := π ∈ P ( a , b , c )

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Denote the collection of a × b plane partitions with entries no bigger than some c > 0 by P ( a , b , c ). For a plane partition π , define � | π | := π i , j . i , j For any triple ( a , b , c ), define � q | π | . M a , b , c ( q ) := π ∈ P ( a , b , c ) In [Rob16], Roby defined a toggling operation η on a plane partition π by changing each entry from bottom to top in each column and from left to right across all columns according to π ′ i , j = min { π i − 1 , j , π i , j − 1 } + max { π i +1 , j , π i , j +1 } − π i , j .

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 3 2 2 0 6 3 1 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 3 2 2 0 6 1 1 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 5 2 2 0 6 1 1 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 5 2 2 0 6 1 0 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 5 5 2 0 6 1 0 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 5 5 2 0 6 1 0 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 5 5 3 0 6 1 0 0 0 0 0 0

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Cyclic Sieving Phenomenon of Plane Partitions Plane Partitions Example Here’s an example of η ( π ) for some plane partition π ∈ P (2 , 3 , 6). 6 6 6 6 5 5 3 0 6 1 0 0 0 0 0 0 Theorem (Hong-Shen-W.) The toggling operation η has order n = a + b, and the triple ( P ( a , b , c ) , η, M a , b , c ( q )) exhibits CSP.

Cyclic Sieving Phenomenon of Plane Partitions and Cluster Duality of Grassmannian Decorated Grassmannian and Decorated Configuration Space Decorated Grassmannian Definition The decorated Grassmannian is defined to be � Mat full rank G r × Mat × G r a ( n ) := SL a a ( n ) := SL a � a , n a , n where superscript × indicates an additional consecutive general position condition.