Simultaneous core partitions for affine Weyl groups Marko Thiel Universit¨atZ¨urich Joint work with Nathan Williams (LACIM, Montr´eal) Marko Thiel Simultaneous core partitions for affine Weyl groups { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups { a-core partitions Outline Marko Thiel Simultaneous core partitions for affine Weyl groups { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups { Sea-action on a-cores Outline { a-core partitions Marko Thiel Simultaneous core partitions for affine Weyl groups { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups { The geometry of the affine symmetric group Sea Outline { a-core partitions { Sea-action on a-cores Marko Thiel Simultaneous core partitions for affine Weyl groups { Simultaneous (a; b)-cores { Other affine Weyl groups { The lattice of cores Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea Marko Thiel Simultaneous core partitions for affine Weyl groups { Other affine Weyl groups { Simultaneous (a; b)-cores Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores Marko Thiel Simultaneous core partitions for affine Weyl groups { Other affine Weyl groups Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores Marko Thiel Simultaneous core partitions for affine Weyl groups Outline { a-core partitions { Sea-action on a-cores { The geometry of the affine symmetric group Sea { The lattice of cores { Simultaneous (a; b)-cores { Other affine Weyl groups Marko Thiel Simultaneous core partitions for affine Weyl groups An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions Marko Thiel Simultaneous core partitions for affine Weyl groups Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Marko Thiel Simultaneous core partitions for affine Weyl groups Ferrers diagram of λ has λi boxes in row i: 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Marko Thiel Simultaneous core partitions for affine Weyl groups 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: Marko Thiel Simultaneous core partitions for affine Weyl groups 6 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: Marko Thiel Simultaneous core partitions for affine Weyl groups 6 4 2 1 1 Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Marko Thiel Simultaneous core partitions for affine Weyl groups Integer partitions An integer partition of size n is a weakly decreasing tuple of positive integers that sum to n. Example: λ = (6; 4; 2; 1; 1) is an integer partition of size 14. Ferrers diagram of λ has λi boxes in row i: 6 7 4 2 1 1 Hook length of a box: # boxes below it + # boxes to the right of it + 1 f g f g Marko Thiel Simultaneous core partitions for affine Weyl groups An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions Marko Thiel Simultaneous core partitions for affine Weyl groups Example: λ = (6; 4; 2; 1; 1) is a 3-core. 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions An a-core is an integer partition with no hook length equal to a. Marko Thiel Simultaneous core partitions for affine Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. Marko Thiel Simultaneous core partitions for affine Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. a-core partitions An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. Marko Thiel Simultaneous core partitions for affine Weyl groups 107 5 4 2 1 8 4 2 1 4 1 2 1 a-core partitions An a-core is an integer partition with no hook length equal to a. Example: λ = (6; 4; 2; 1; 1) is a 3-core. Originally introduced by Nakayama in the study of the representation theory of the symmetric group in prime characteristic. Marko Thiel Simultaneous core partitions for affine Weyl groups The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape. Adding/Removing boxes Marko Thiel Simultaneous core partitions for affine Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − Marko Thiel Simultaneous core partitions for affine Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 removable box: Can remove this box and still have a Ferrers shape. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − Marko Thiel Simultaneous core partitions for affine Weyl groups 0 1 2 0 1 2 2 0 1 2 1 2 3-residues of a 3-core 0 2 Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − removable box: Can remove this box and still have a Ferrers shape. Marko Thiel Simultaneous core partitions for affine Weyl groups addable box: Can add this box and still have a Ferrers shape. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 2 0 1 2 1 2 0 2 removable box: Can remove this box and still have a Ferrers shape. Marko Thiel Simultaneous core partitions for affine Weyl groups Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 2 0 1 2 1 2 0 2 removable box: Can remove this box and still have a Ferrers shape. addable box: Can add this box and still have a Ferrers shape. Marko Thiel Simultaneous core partitions for affine Weyl groups Note: There are never both addable and removable boxes of the same residue. Adding/Removing boxes The a-residue of a box in row i and column j is r(i; j) = j i mod a: − 0 1 2 0 1 2 0 2 0 1 2 0 1 2 0 0 1 2 1 removable box: Can remove this box and still have a Ferrers shape.