Self-conjugate core partitions: Its storytime! Christopher R. H. - PowerPoint PPT Presentation

Self-conjugate core partitions: It’s storytime! Christopher R. H. Hanusa Queens College, CUNY Joint work with Rishi Nath, York College, CUNY people.qc.cuny.edu/chanusa > Talks

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition The Young diagram of λ = ( λ 1 , . . . , λ k ) has λ i boxes in row i . Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition 10 9 6 5 2 1 7 6 3 2 6 5 2 1 3 2 2 1 The Young diagram of λ = ( λ 1 , . . . , λ k ) has λ i boxes in row i . The hook length of a box = # boxes below + # boxes to right + box λ is a t -core if no boxes have hook length t . Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition 10 9 6 5 2 1 7 6 3 2 6 5 2 1 3 2 2 1 The Young diagram of λ = ( λ 1 , . . . , λ k ) has λ i boxes in row i . The hook length of a box = # boxes below + # boxes to right + box λ is a t -core if no boxes have hook length t . Example: Mr. Core is not 3-, 5-, 6-core; is a 4-, 8-, 11-core. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition Coxeter groups: t -cores biject with 10 9 6 5 2 1 min. wt. coset reps 7 6 3 2 in � 6 5 2 1 A t / A t . (action) 3 2 � � 4, � 3,7,10 � window 2 1 notation s 1 s 0 s 2 s 3 s 1 s 0 s 2 s 3 s 1 s 0 reduced abacus expression diagram elements of � � A A bounded core partition partition root lattice point � � 1,2,1, � 2 � The Young diagram of λ = ( λ 1 , . . . , λ k ) has λ i boxes in row i . The hook length of a box = # boxes below + # boxes to right + box λ is a t -core if no boxes have hook length t . Example: Mr. Core is not 3-, 5-, 6-core; is a 4-, 8-, 11-core. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition Coxeter groups: Representation Theory: t -cores biject with 10 9 6 5 2 1 t -cores label the min. wt. coset reps 7 6 3 2 t -blocks of in � 6 5 2 1 A t / A t . (action) 3 2 irreducible � � 4, � 3,7,10 � window 2 1 characters notation s 1 s 0 s 2 s 3 s 1 s 0 s 2 s 3 s 1 s 0 reduced abacus expression diagram of S n . elements of � � A A bounded core partition partition Mock theta root lattice point functions � � 1,2,1, � 2 � The Young diagram of λ = ( λ 1 , . . . , λ k ) has λ i boxes in row i . The hook length of a box = # boxes below + # boxes to right + box λ is a t -core if no boxes have hook length t . Example: Mr. Core is not 3-, 5-, 6-core; is a 4-, 8-, 11-core. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mr. Core Partition Coxeter groups: Representation Theory: t -cores biject with 10 9 6 5 2 1 t -cores label the min. wt. coset reps 7 6 3 2 t -blocks of in � 6 5 2 1 A t / A t . (action) 3 2 irreducible � � 4, � 3,7,10 � window 2 1 characters notation s 1 s 0 s 2 s 3 s 1 s 0 s 2 s 3 s 1 s 0 reduced abacus expression diagram of S n . elements of � � A A bounded core Let c t ( n ) be the number partition partition Mock theta root lattice of t -core partitions of n . point functions � � 1,2,1, � 2 � The Young diagram of λ = ( λ 1 , . . . , λ k ) has λ i boxes in row i . The hook length of a box = # boxes below + # boxes to right + box λ is a t -core if no boxes have hook length t . Example: Mr. Core is not 3-, 5-, 6-core; is a 4-, 8-, 11-core. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 1 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mrs. Core Partition Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 2 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mrs. Core Partition A partition is self-conjugate if it is symmetric about its main diagonal. In this talk: Understanding self-conjugate core partitions. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 2 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mrs. Core Partition 13 9 7 5 3 2 1 9 5 3 1 7 3 1 5 1 3 2 1 A partition is self-conjugate if it is symmetric about its main diagonal. In this talk: Understanding self-conjugate core partitions. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 2 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mrs. Core Partition Coxeter groups: s-c t -cores biject 13 9 7 5 3 2 1 9 5 3 1 with min. wt. coset 7 3 1 reps in � C t / C t . 5 1 (Hanusa, Jones ’12) 3 2 � � 11, � 9, � 1,8,16,18 � 1 window s 0 s 1 s 0 s 3 s 2 s 1 s 0 s 2 s 3 notation s 2 s 1 s 0 s 2 s 3 s 2 s 1 s 0 reduced abacus expression diagram elements of � � C C bounded core partition partition root lattice point � 1,2, � 2 � A partition is self-conjugate if it is symmetric about its main diagonal. In this talk: Understanding self-conjugate core partitions. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 2 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mrs. Core Partition Coxeter groups: Representation s-c t -cores biject Theory: 13 9 7 5 3 2 1 9 5 3 1 with min. wt. coset s-c t -cores label 7 3 1 reps in � C t / C t . defect zero 5 1 (Hanusa, Jones ’12) 3 t -blocks of A n 2 that arise from � � 11, � 9, � 1,8,16,18 � 1 window s 0 s 1 s 0 s 3 s 2 s 1 s 0 s 2 s 3 splitting t -blocks notation s 2 s 1 s 0 s 2 s 3 s 2 s 1 s 0 reduced abacus expression diagram of S n . elements of � � C C bounded core partition partition root lattice point (Ask Rishi) � 1,2, � 2 � A partition is self-conjugate if it is symmetric about its main diagonal. In this talk: Understanding self-conjugate core partitions. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 2 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Meet Mrs. Core Partition Coxeter groups: Representation s-c t -cores biject Theory: 13 9 7 5 3 2 1 9 5 3 1 with min. wt. coset s-c t -cores label 7 3 1 reps in � C t / C t . defect zero 5 1 (Hanusa, Jones ’12) 3 t -blocks of A n 2 that arise from � � 11, � 9, � 1,8,16,18 � 1 window s 0 s 1 s 0 s 3 s 2 s 1 s 0 s 2 s 3 splitting t -blocks notation s 2 s 1 s 0 s 2 s 3 s 2 s 1 s 0 reduced abacus expression diagram of S n . elements of Let sc t ( n ) be the number � � C C bounded core of self-conjugate t -core partition partition root lattice point partitions of n . (Ask Rishi) � 1,2, � 2 � A partition is self-conjugate if it is symmetric about its main diagonal. In this talk: Understanding self-conjugate core partitions. Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 2 / 14

Meet our actors: Core Partitions Act I: Large cores Act II: Small cores Coming attractions Beauty contest Core partitions Self-conjugate core partitions Self-conjugate core partitions Combinatorial and Additive Number Theory 2012 Christopher R. H. Hanusa Queens College, CUNY May 24, 2012 3 / 14