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Diagrams of Affine Permutations and Their Labellings Diagrams of Affine Permutations and Their Labellings Taedong Yun Oracle June 23, 2014 Taedong Yun Permutation Diagrams and Labellings June 23, 2014 1 / 36 Contents Based on the joint work with Hwanchul Yoo (KIAS). Taedong Yun Permutation Diagrams and Labellings June 23, 2014 2 / 36 Affine Balanced Labellings Section 1 Affine Balanced Labellings Taedong Yun Permutation Diagrams and Labellings June 23, 2014 3 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Permutations A permutation is a bijection w : f1; 2;:::; ng ! f1; 2;:::; ng. Σn := the symmetric group, group of (finite) permutations of size n. Σn is generated by 1 the simple reflections s1;:::; sn−1 (si is the permutation which interchanges the pair (i; i + 1)) 2 and the following relations 2 si = 1 for all i si si+1si = si+1si si+1 for all i si sj = sj si for ji − jj ≥ 2: Taedong Yun Permutation Diagrams and Labellings June 23, 2014 4 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Affine Permutations An affine permutation of period n is a bijection w : Z ! Z such that w(i + n) = w(i) + n 8i 2 Z and w(1) + w(2) + ··· + w(n) = n(n + 1)=2. Σen := the affine symmetric group, group of affine permutations of period n. Σen is generated by 1 the simple reflections s0; s1;:::; sn−1 (si interchanges the pairs (i+rn; i + 1+rn) 8r 2 Z:) 2 and the following relations 2 si = 1 for all i si si+1si = si+1si si+1 for all i si sj = sj si for i; j not adjacent, i 6= j: where the indices are taken modulo n. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 5 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Affine Permutations Example Σe3 is generated by fs0; s1; s2g. ··· -1 0 1 2 3 4 5 ··· id ··· -1 0 1 2 3 4 5 ··· s2 ··· 0 -1 132 46 ··· s2s0 ··· 01 -134 26 ··· s2s0s1 ··· -41 3 -14 62 ··· s2s0s1s0 ··· -43 1 -16 42 ··· w = s2s0s1s0 2 Σe3 w = [1; −1; 6] : window notation Taedong Yun Permutation Diagrams and Labellings June 23, 2014 6 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Affine Permutations Remark A permutation w = [w1;:::; wn] can be viewed as an affine permutation [w1;:::; wn] with period n, written in window notation. This corresponds to the natural embedding Σn ,−! Σen, si 7−! si . Taedong Yun Permutation Diagrams and Labellings June 23, 2014 7 / 36 Fact The length of an affine permutation w is the number of inversions (i; j) with 1 ≤ i ≤ n. `(w) = jf(i; j) j i < j; w(i) > w(j); 1 ≤ i ≤ ngj: Affine Balanced Labellings Affine Permutations and Balanced Labellings Reduced Words A reduced decomposition of w is a decomposition w = si1 ··· si` where ` is the minimal number for which such a decomposition exists. That minimal ` = `(w) is the length of w. The word i1i2 ··· i` is called a reduced word of w. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 8 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Reduced Words A reduced decomposition of w is a decomposition w = si1 ··· si` where ` is the minimal number for which such a decomposition exists. That minimal ` = `(w) is the length of w. The word i1i2 ··· i` is called a reduced word of w. Fact The length of an affine permutation w is the number of inversions (i; j) with 1 ≤ i ≤ n. `(w) = jf(i; j) j i < j; w(i) > w(j); 1 ≤ i ≤ ngj: Taedong Yun Permutation Diagrams and Labellings June 23, 2014 8 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Diagrams The affine permutation diagram of w 2 Σen is the set D(w) = f(i; w(j)) j i < j; w(i) > w(j)g ⊆ Z × Z: When w is a finite permutation, D(w) consists of infinite number of identical copies of the Rothe diagram of w diagonally. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 9 / 36 fundamental window [D(w)] := ([1; n] × Z) \ D(w). cell = f(i + rn; j + rn) j r 2 Zg =: (i; j). `(w) = #(cells) = j[D(w)]j Affine Balanced Labellings Affine Permutations and Balanced Labellings Diagrams 1 6 1 6 Figure: diagram of w = [2; 5; 0; 7; 3; 4] 2 Σe6 Taedong Yun Permutation Diagrams and Labellings June 23, 2014 10 / 36 Affine Balanced Labellings Affine Permutations and Balanced Labellings Diagrams 1 6 1 fundamental window [D(w)] := ([1; n] × Z) \ D(w). cell = f(i + rn j + rn) j r 2 g 6 ; Z =: (i; j). `(w) = #(cells) = j[D(w)]j Figure: diagram of w = [2; 5; 0; 7; 3; 4] 2 Σe6 Taedong Yun Permutation Diagrams and Labellings June 23, 2014 10 / 36 4 7 4 2 5 3 1 7 Hook Hij := (fig × Z≥j ) [ (Z≥i × fjg) Balanced hook: If one rearranges the labels in the hook so that they weakly increase from right to left and from top to bottom, then the corner label remains unchanged. Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Hook Labelling of a diagram = labelling of the cells with positive integers. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 11 / 36 4 7 4 2 5 3 1 7 Balanced hook: If one rearranges the labels in the hook so that they weakly increase from right to left and from top to bottom, then the corner label remains unchanged. Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Hook Labelling of a diagram = labelling of the cells with positive integers. Hook Hij := (fig × Z≥j ) [ (Z≥i × fjg) Taedong Yun Permutation Diagrams and Labellings June 23, 2014 11 / 36 4 7 4 2 5 3 1 7 Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Hook Labelling of a diagram = labelling of the cells with positive integers. Hook Hij := (fig × Z≥j ) [ (Z≥i × fjg) Balanced hook: If one rearranges the labels in the hook so that they weakly increase from right to left and from top to bottom, then the corner label remains unchanged. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 11 / 36 4 7 4 2 5 3 1 7 Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Hook Labelling of a diagram = labelling of the cells with positive integers. 4 4 3 2 1 5 7 7 Hook Hij := (fig × Z≥j ) [ (Z≥i × fjg) Balanced hook: If one rearranges the labels in the hook so that they weakly increase from right to left and from top to bottom, then the corner label remains unchanged. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 11 / 36 3 4 1 2 3 4 1 column-strict if no column contains two boxes with equal 2 labels injective if each of the labels 3 f1; 2; : : : ; `g appears exactly once in [D]. Figure: injective balanced labelling Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Labellings Definition (Fomin-Greene-Reiner-Shimozono 1997) A labelling of a diagram D is called: balanced if every hook is balanced Taedong Yun Permutation Diagrams and Labellings June 23, 2014 12 / 36 3 4 1 2 3 4 1 2 injective if each of the labels 3 f1; 2; : : : ; `g appears exactly once in [D]. Figure: injective balanced labelling Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Labellings Definition (Fomin-Greene-Reiner-Shimozono 1997) A labelling of a diagram D is called: balanced if every hook is balanced column-strict if no column contains two boxes with equal labels Taedong Yun Permutation Diagrams and Labellings June 23, 2014 12 / 36 3 4 1 2 3 4 1 2 3 Affine Balanced Labellings Affine Permutations and Balanced Labellings Balanced Labellings Definition (Fomin-Greene-Reiner-Shimozono 1997) A labelling of a diagram D is called: balanced if every hook is balanced column-strict if no column contains two boxes with equal labels injective if each of the labels f1; 2; : : : ; `g appears exactly once in [D]. Figure: injective balanced labelling Taedong Yun Permutation Diagrams and Labellings June 23, 2014 12 / 36 1 1 2 1 1 2 1 1 2 s2 s2s0 1 2 3 4 1 3 1 2 2 3 4 1 3 1 2 2 3 s2s0s1 s2s0s1s0 Affine Balanced Labellings Reduced Words and Canonical Labellings Canonical Labellings Let a = a1a2 ··· a` be a reduced word of w 2 Σen, i.e. w = sa1 sa2 ··· sa` . We define the canonical labelling Ta of a. e.g. w = [1; −1; 6] = s2s0s1s0. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 13 / 36 1 1 2 1 1 2 1 1 2 s2 s2s0 1 2 3 4 1 3 1 2 2 3 4 1 3 1 2 2 3 s2s0s1 s2s0s1s0 Affine Balanced Labellings Reduced Words and Canonical Labellings Canonical Labellings Let a = a1a2 ··· a` be a reduced word of w 2 Σen, i.e. w = sa1 sa2 ··· sa` . We define the canonical labelling Ta of a. e.g. w = [1; −1; 6] = s2s0s1s0. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 13 / 36 Affine Balanced Labellings Reduced Words and Canonical Labellings Canonical Labellings Theorem (Yoo-Y. 2012, FGRS 1997 for finite case) For all w 2 Σen, the map a 7−! Ta (the canonical labelling of a) is a bijection freduced words of wg −! finjective balanced labellings of D(w)g. Proof. Find the inverse map. Taedong Yun Permutation Diagrams and Labellings June 23, 2014 14 / 36 Symmetric Functions Section 2 Symmetric Functions Taedong Yun Permutation Diagrams and Labellings June 23, 2014 15 / 36 Symmetric Functions Affine Stanley Symmetric Functions Stanley Symmetric Functions v 2 Σn is called decreasing if it has a decreasing reduced word.
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