Equality of Schur Supports of Ribbons Other Results - PowerPoint PPT Presentation

Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran Preliminaries Main Results Equality of Schur Supports of Ribbons Other Results Acknowledgements References Marisa Gaetz, Will Hardt, Shruthi Sridhar, Anh Quoc Tran Research work from UMN Twin Cities REU 2017 August 2, 2017

Overview Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran 1 Preliminaries Preliminaries Main Results Other Results Acknowledgements 2 Main Results References 3 Other Results 4 Acknowledgements

Schur Functions Equality of Schur Supports of Ribbons Example/Definition ( Young Diagram & Semistandard Young Tableau) Gaetz, Hardt, Sridhar, Tran Partition λ = (4 , 3 , 2) Preliminaries 1 1 2 3 Main Results 2 2 3 Other Results 3 4 Acknowledgements Young Diagram SSYT References Definition The Schur function s λ of a partition λ is x T = � � x t 1 1 x t 2 2 x t 3 s λ ( x 1 , x 2 , x 3 , . . . ) = 3 · · · T : SSYT T of shape λ where t i is the number of occurrences of i in T .

Skew Schur Functions Example/Definition (Skew Shape) Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran Preliminaries λ = µ = λ/µ = Main Results Other Results Acknowledgements References

Skew Schur Functions Example/Definition (Skew Shape) Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran Preliminaries λ = µ = λ/µ = Main Results Skew Schur functions are: Other Results defined analogously to straight Schur functions. Acknowledgements References Schur-positive, meaning � c ν s λ/µ = λ,µ s ν ν where ν is a straight partition, and c λ µ,ν ≥ 0.

Skew Schur Functions Example/Definition (Skew Shape) Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran Preliminaries λ = µ = λ/µ = Main Results Skew Schur functions are: Other Results defined analogously to straight Schur functions. Acknowledgements References Schur-positive, meaning � c ν s λ/µ = λ,µ s ν ν where ν is a straight partition, and c λ µ,ν ≥ 0. Definition The Schur support of a skew shape λ/µ , denoted [ λ/µ ], is defined as [ λ/µ ] = { ν : c λ µ,ν > 0 } .

Ribbons Equality of Schur Supports of Ribbons Gaetz, Hardt, A ribbon is a skew shape which does not contain a 2 × 2 subdiagram. Sridhar, Tran Example Preliminaries Main Results Ribbon : Non-ribbon: Other Results Acknowledgements References Given a sequence of integers, there’s a unique ribbon with that sequence of row lengths. Thus, ribbons are uniquely determined by compositions of n (the total boxes). The above example can be denoted as the ribbon (3 , 2 , 3)

Littlewood-Richardson Rule This is a rule to check if a particular straight young diagram is Equality of Schur Supports of present in the support. Ribbons Gaetz, Hardt, Sridhar, Tran Formal Definition Preliminaries Let D be a skew shape. A partition λ = ( λ 1 , . . . , λ m ) is in the Main Results support of s D iff there is a valid LR-filling of D with content λ . Other Results Acknowledgements A filling of D is an LR-filling if: References The tableau is semistandard. Every initial reverse reading word is Yamanouchi : # i ’s ≥ #( i + 1)’s

Littlewood-Richardson Rule This is a rule to check if a particular straight young diagram is Equality of Schur Supports of present in the support. Ribbons Gaetz, Hardt, Sridhar, Tran Formal Definition Preliminaries Let D be a skew shape. A partition λ = ( λ 1 , . . . , λ m ) is in the Main Results support of s D iff there is a valid LR-filling of D with content λ . Other Results Acknowledgements A filling of D is an LR-filling if: References The tableau is semistandard. Every initial reverse reading word is Yamanouchi : # i ’s ≥ #( i + 1)’s Example/Definition (Yamanouchi Property) Reverse Reading Word: 1,1,1,2,2,3,2 1 1 2 2 1 This is Yamanouchi because there are at least 2 3 as many 1’s as 2’s and as many 2’s as 3’s at every stage.

Littlewood Richardson Rule Equality of Schur Example of LR-rule Supports of Ribbons Gaetz, Hardt, Reverse Reading Word: 1,1,2,2,1,3,2 1 1 Sridhar, Tran 1 2 2 This is Yamanouchi and semistandard, hence is Preliminaries 2 3 a valid LR-filling Main Results Other Results The content of the filling is (3,3,1), thus (3,3,1) is in the support of Acknowledgements the ribbon: (2 , 3 , 2). References

Littlewood Richardson Rule Equality of Schur Example of LR-rule Supports of Ribbons Gaetz, Hardt, Reverse Reading Word: 1,1,2,2,1,3,2 1 1 Sridhar, Tran 1 2 2 This is Yamanouchi and semistandard, hence is Preliminaries 2 3 a valid LR-filling Main Results Other Results The content of the filling is (3,3,1), thus (3,3,1) is in the support of Acknowledgements the ribbon: (2 , 3 , 2). References Proposition Let α = (1 , α 2 , α 3 ) be a ribbon. Then, α ′ = ( α 2 , 1 , α 3 ) and α don’t have the same support. Proof. When the row of length 1 1 1 1 is in the middle, there is no 1 2 2 LR-filling with just 1’s and 2’s. 1 2 ∗ 3 α ′ = (2 , 1 , 2) α = (1 , 2 , 2)

Ribbons with Equal Support Equality of Schur Supports of [McNamara (2008)] gives that 2 ribbons can have the same support Ribbons only if one is a permutation of the rows of the other. Gaetz, Hardt, Sridhar, Tran Definition Preliminaries Main Results Let α = ( α 1 , α 2 , . . . , α m ) be a ribbon. We use α π to denote a ribbon Other Results formed by applying the permutation π ∈ S m to the row lengths of α . Acknowledgements References

Ribbons with Equal Support Equality of Schur Supports of [McNamara (2008)] gives that 2 ribbons can have the same support Ribbons only if one is a permutation of the rows of the other. Gaetz, Hardt, Sridhar, Tran Definition Preliminaries Main Results Let α = ( α 1 , α 2 , . . . , α m ) be a ribbon. We use α π to denote a ribbon Other Results formed by applying the permutation π ∈ S m to the row lengths of α . Acknowledgements References α = α (2 3) = α (1 2) = α (1 3 2) = α (1 2 3) = α (1 3) =

Ribbons with Equal Support Equality of Schur Supports of [McNamara (2008)] gives that 2 ribbons can have the same support Ribbons only if one is a permutation of the rows of the other. Gaetz, Hardt, Sridhar, Tran Definition Preliminaries Main Results Let α = ( α 1 , α 2 , . . . , α m ) be a ribbon. We use α π to denote a ribbon Other Results formed by applying the permutation π ∈ S m to the row lengths of α . Acknowledgements References α = α (2 3) = α (1 2) = α (1 3 2) = α (1 2 3) = α (1 3) = Definition A ribbon α = ( α 1 , α 2 , . . . , α m ) is said to have full equivalence class if for any permutation π ∈ S m , we have [ α ] = [ α π ].

Trivial Cases Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran Proposition Preliminaries If a ribbon α = ( α 1 , α 2 , . . . , α m ) has k rows of length 1, where Main Results 1 ≤ k < m, then α does not have full equivalence class. Other Results Acknowledgements References

Trivial Cases Equality of Schur Supports of Ribbons Gaetz, Hardt, Sridhar, Tran Proposition Preliminaries If a ribbon α = ( α 1 , α 2 , . . . , α m ) has k rows of length 1, where Main Results 1 ≤ k < m, then α does not have full equivalence class. Other Results Acknowledgements References Remark It is well known that rotating a ribbon by 180 ◦ preserves its support. It follows trivially that any ribbon with only two rows has full equivalence class. For the rest of the presentation, we consider only ribbons with more than two rows and with no rows of length 1.

Sufficient Condition for Full E.C Equality of Schur Supports of Ribbons Theorem (S, G, H, Tran, ’17) Gaetz, Hardt, Sridhar, Tran Let α = ( α 1 , α 2 , . . . , α m ) be a ribbon such that any subset of size Preliminaries three of { α i } satisfies the strict triangle inequality ( α i < α j + α k ). Main Results Then α has full equivalence class. Other Results Acknowledgements Proof Idea: Given a ribbon with an LR-filling, show how to swap References two adjacent row lengths while preserving the content, Yamanouchi property, and semistandardness of the filling.

Sufficient Condition for Full E.C Equality of Schur Supports of Ribbons Theorem (S, G, H, Tran, ’17) Gaetz, Hardt, Sridhar, Tran Let α = ( α 1 , α 2 , . . . , α m ) be a ribbon such that any subset of size Preliminaries three of { α i } satisfies the strict triangle inequality ( α i < α j + α k ). Main Results Then α has full equivalence class. Other Results Acknowledgements Proof Idea: Given a ribbon with an LR-filling, show how to swap References two adjacent row lengths while preserving the content, Yamanouchi property, and semistandardness of the filling. Proof Sketch: 1 Use the R -matrix algorithm (described on the next slide) to swap adjacent row lengths while preserving content and the Yamanouchi property. 2 Show how to adjust the resulting filling to be semistandard.

R -Matrix Algorithm Equality of Schur Algorithm [Inoue et al. (2012), Section 2.2.3] Supports of Ribbons Gaetz, Hardt, 1 Represent the rows to be swapped as box-ball systems. Sridhar, Tran Preliminaries Main Results Other Results Acknowledgements References � � ⊗ = ⊗ R 1 3 3 4 7 1 3 5 1 4 7 1 3 3 3 5