Standard Young Tableaux Old and New Ron Adin and Yuval Roichman - PowerPoint PPT Presentation

Standard Young Tableaux – Old and New Ron Adin and Yuval Roichman Department of Mathematics Bar-Ilan University Workshop on Group Theory in Memory of David Chillag Technion, Haifa, Oct. ’14 1 2 4 1 2 4 3 5 7 3 5 7 6 8 6 8 9 9

Classical Still Classical Non-Classical David Chillag

Classical Still Classical Non-Classical Abstract More than a hundred years ago, Frobenius and Young based the emerging representation theory of the symmetric group on the combinatorial objects now called Standard Young Tableaux (SYT). Many important features of these classical objects have since been discovered, including some surprising interpretations and the celebrated hook length formula for their number. In recent years, SYT of non-classical shapes have come up in research and were shown to have, in many cases, surprisingly nice enumeration formulas. The talk will present some gems from the study of SYT over the years, based on a recent survey paper. No prior acquaintance assumed.

Classical Still Classical Non-Classical Founders

Classical Still Classical Non-Classical Founders A. Young

Classical Still Classical Non-Classical Founders A. Young F. G. Frobenius

Classical Still Classical Non-Classical Founders A. Young F. G. Frobenius P. A. MacMahon

Classical Still Classical Non-Classical Classical

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides.

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 1

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 2 1

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 2 3 1

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 4 2 3 1

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 5 4 2 3 1

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 5 4 2 3 1 Rotate:

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 5 4 2 3 1 Rotate: 1 2 → 3 5 4

Classical Still Classical Non-Classical Introduction Consider throwing balls labeled 1 , 2 , . . . , n into a V-shaped bin with perpendicular sides. 5 4 2 3 1 Rotate: 1 2 1 2 → 3 5 → 3 5 4 4

Classical Still Classical Non-Classical Diagrams and Tableaux

Classical Still Classical Non-Classical Diagrams and Tableaux ← → partition diagram/shape λ = (4 , 3 , 1) ⊢ 8 [ λ ] =

Classical Still Classical Non-Classical Diagrams and Tableaux ← → partition diagram/shape λ = (4 , 3 , 1) ⊢ 8 [ λ ] = Standard Young Tableau (SYT): 1 2 5 8 ∈ SYT(4 , 3 , 1) . T = 3 4 6 7 Entries increase along rows and columns

Classical Still Classical Non-Classical Conventions 1 2 4 5 4 2 3 3 5 3 5 1 4 1 2 English Russian French

Classical Still Classical Non-Classical Number of SYT

Classical Still Classical Non-Classical Number of SYT f λ = # SYT( λ )

Classical Still Classical Non-Classical Number of SYT f λ = # SYT( λ ) 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

Classical Still Classical Non-Classical Number of SYT f λ = # SYT( λ ) 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 f λ = 5 λ = (3 , 2) ,

Classical Still Classical Non-Classical SYT and S n Representations S n = the symmetric group on n letters

Classical Still Classical Non-Classical SYT and S n Representations S n = the symmetric group on n letters χ λ λ ← → partition of n irreducible character of S n

Classical Still Classical Non-Classical SYT and S n Representations S n = the symmetric group on n letters χ λ λ ← → partition of n irreducible character of S n SYT( λ ) ← → basis of representation space

Classical Still Classical Non-Classical SYT and S n Representations S n = the symmetric group on n letters χ λ λ ← → partition of n irreducible character of S n SYT( λ ) ← → basis of representation space f λ χ λ ( id ) =

Classical Still Classical Non-Classical SYT and S n Representations S n = the symmetric group on n letters χ λ λ ← → partition of n irreducible character of S n SYT( λ ) ← → basis of representation space f λ χ λ ( id ) = Corollary: ( f λ ) 2 = n ! � λ ⊢ n

Classical Still Classical Non-Classical RS(K) Correspondence [Robinson, Schensted (, Knuth)]

Classical Still Classical Non-Classical RS(K) Correspondence [Robinson, Schensted (, Knuth)] π ← → ( P , Q ) permutation pair of SYT of the same shape

Classical Still Classical Non-Classical RS(K) Correspondence [Robinson, Schensted (, Knuth)] π ← → ( P , Q ) permutation pair of SYT of the same shape   1 3 5 7 1 3 4 7   4236517 ← → , 2 6 2 5     4 6

Classical Still Classical Non-Classical RS(K) Correspondence [Robinson, Schensted (, Knuth)] π ← → ( P , Q ) permutation pair of SYT of the same shape   1 3 5 7 1 3 4 7   4236517 ← → , 2 6 2 5     4 6 Corollary: ( f λ ) 2 = n ! � λ ⊢ n

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams.

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ →

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → →

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → → →

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → → → →

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → → → → →

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → → → → → The Young lattice consists of all partitions (diagrams), of all sizes, ordered by inclusion.

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → → → → → The Young lattice consists of all partitions (diagrams), of all sizes, ordered by inclusion. SYT( λ ) ← → maximal chains in the Young lattice from ∅ to λ

Classical Still Classical Non-Classical Interpretation: The Young Lattice A SYT describes a growth process of diagrams. For example, 1 2 5 3 4 corresponds to the process ∅ → → → → → The Young lattice consists of all partitions (diagrams), of all sizes, ordered by inclusion. SYT( λ ) ← → maximal chains in the Young lattice from ∅ to λ The number of such maximal chains is therefore f λ .

Classical Still Classical Non-Classical Interpretation: Lattice Paths Each SYT of shape λ = ( λ 1 , . . . , λ t ) corresponds to a lattice path in R t , from the origin 0 to the point λ , where in each step exactly one of the coordinates changes (by adding 1), while staying within the region { ( x 1 , . . . , x t ) ∈ R t | x 1 ≥ . . . ≥ x t ≥ 0 } .

Classical Still Classical Non-Classical Interpretation: Lattice Paths Each SYT of shape λ = ( λ 1 , . . . , λ t ) corresponds to a lattice path in R t , from the origin 0 to the point λ , where in each step exactly one of the coordinates changes (by adding 1), while staying within the region { ( x 1 , . . . , x t ) ∈ R t | x 1 ≥ . . . ≥ x t ≥ 0 } . x 2 x 1

Classical Still Classical Non-Classical Interpretation: Lattice Paths Each SYT of shape λ = ( λ 1 , . . . , λ t ) corresponds to a lattice path in R t , from the origin 0 to the point λ , where in each step exactly one of the coordinates changes (by adding 1), while staying within the region { ( x 1 , . . . , x t ) ∈ R t | x 1 ≥ . . . ≥ x t ≥ 0 } . x 2 1 x 1