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On (rational) Shi tableaux Robin Sulzgruber 78 th S eminaire - PowerPoint PPT Presentation

Feb 09, 2023 •205 likes •1.36k views

On (rational) Shi tableaux Robin Sulzgruber 78 th S eminaire Lotharingien de Combinatoire March 26 th 29 th 2017 Ottrott France Robin Sulzgruber On (rational) Shi tableaux March 2017 1 / 33 Setting the stage Robin Sulzgruber On

  1. The height of a hyperplane Definition Define the height of a hyperplane H α, k as | ht( α ) − hk | . 13 10 7 4 1 2 5 8 5 13 2 11 [1 , 2 , 3] 1 10 4 14 7 14 11 8 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 8 / 33

  2. Shi alcoves Robin Sulzgruber On (rational) Shi tableaux March 2017 9 / 33

  3. Shi alcoves Theorem (Shi 1987, Athanasiadis 2005, Thiel 2015) The regions of the m -Shi arrangement are in bijection with alcoves whose floors have height less than mh + 1. Robin Sulzgruber On (rational) Shi tableaux March 2017 9 / 33

  4. Shi alcoves Theorem (Shi 1987, Athanasiadis 2005, Thiel 2015) The regions of the m -Shi arrangement are in bijection with alcoves whose floors have height less than mh + 1. 13 10 7 4 1 2 5 8 8 5 [1 , 0 , 5] 13 [ − 2 , 5 , 3] [0 , 1 , 5] [2 , 0 , 4] [4 , 2 , 0] 2 [2 , 1 , 3] [0 , 2 , 4] 11 [2 , 3 , 1] [1 , 2 , 3] [0 , 4 , 2] 1 [3 , 2 , 1] [1 , 3 , 2] [ − 1 , 4 , 3] 10 [3 , 1 , 2] [ − 1 , 3 , 4] 4 14 [1 , − 1 , 6] 7 7 10 14 11 8 5 2 1 4 Robin Sulzgruber On (rational) Shi tableaux March 2017 9 / 33

  5. Robin Sulzgruber On (rational) Shi tableaux March 2017 10 / 33

  6. 19 16 13 10 7 4 1 2 5 14 8 11 [1 , − 3 , 8] 19 8 [5 , 1 , 0] [ − 3 , 5 , 4] 11 [ − 5 , 8 , 3] [0 , − 2 , 8] [5 , 0 , 1] [1 , 5 , 0] [ − 3 , 4 , 5] [2 , − 3 , 7] [7 , 2 , − 3] 5 [0 , 5 , 1] [1 , 0 , 5] [2 , 4 , 0] 16 [5 , 3 , − 2] [ − 2 , 5 , 3] [0 , 1 , 5] [2 , 0 , 4] [4 , 2 , 0] [ − 3 , 7 , 2] 2 [ − 2 , 3 , 5] [2 , 1 , 3] [0 , 2 , 4] [4 , 0 , 2] 14 [3 , − 2 , 5] [2 , 3 , 1] [1 , 2 , 3] [0 , 4 , 2] [ − 1 , 0 , 7] 1 [2 , − 2 , 6] [3 , 2 , 1] [1 , 3 , 2] [ − 1 , 4 , 3] [0 , − 1 , 7] 13 [ − 2 , 2 , 6] [3 , 1 , 2] [ − 1 , 3 , 4] [4 , − 1 , 3] 4 [6 , 2 , − 2] [ − 2 , 6 , 2] [ − 1 , 1 , 6] [3 , − 1 , 4] [4 , 3 , − 1] [ − 4 , 7 , 3] 17 [ − 1 , 6 , 1] [1 , − 1 , 6] [3 , 4 , − 1] 7 10 [6 , 1 , − 1] [ − 4 , 6 , 4] 10 20 [1 , − 4 , 9] 13 7 16 20 17 14 11 8 5 2 1 4 Robin Sulzgruber On (rational) Shi tableaux March 2017 10 / 33

  7. Inverse Shi alcoves Robin Sulzgruber On (rational) Shi tableaux March 2017 11 / 33

  8. Inverse Shi alcoves Theorem (Fishel, Vazirani 2010) The regions of the m -Shi arrangement are in bijection with the alcoves inside the simplex bounded by the hyperplanes of height mh + 1. Robin Sulzgruber On (rational) Shi tableaux March 2017 11 / 33

  9. Inverse Shi alcoves Theorem (Fishel, Vazirani 2010) The regions of the m -Shi arrangement are in bijection with the alcoves inside the simplex bounded by the hyperplanes of height mh + 1. 16 13 10 7 4 1 2 5 11 16 8 8 [1 , 5 , 0] 5 [1 , 0 , 5] 13 [0 , 1 , 5] [2 , 0 , 4] 2 [2 , 1 , 3] [0 , 2 , 4] 11 [2 , 3 , 1] [1 , 2 , 3] [0 , 4 , 2] 1 [3 , 2 , 1] [1 , 3 , 2] [ − 1 , 4 , 3] 10 [ − 2 , 2 , 6] [3 , 1 , 2] [ − 1 , 3 , 4] [4 , − 1 , 3] 4 14 7 14 11 8 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 11 / 33

  10. Robin Sulzgruber On (rational) Shi tableaux March 2017 12 / 33

  11. 22 19 16 13 10 7 4 1 2 5 17 22 14 8 [1 , 8 , − 3] 11 [1 , − 3 , 8] 19 [ − 3 , 1 , 8] [5 , − 3 , 4] 8 [5 , 1 , 0] [ − 3 , 5 , 4] 11 [5 , 0 , 1] [1 , 5 , 0] [ − 3 , 4 , 5] 5 [0 , 5 , 1] [1 , 0 , 5] [2 , 4 , 0] 16 [ − 2 , 5 , 3] [0 , 1 , 5] [2 , 0 , 4] [4 , 2 , 0] 2 [ − 2 , 3 , 5] [2 , 1 , 3] [0 , 2 , 4] [4 , 0 , 2] 14 [3 , − 2 , 5] [2 , 3 , 1] [1 , 2 , 3] [0 , 4 , 2] [ − 1 , 0 , 7] 1 [2 , − 2 , 6] [3 , 2 , 1] [1 , 3 , 2] [ − 1 , 4 , 3] [0 , − 1 , 7] 13 [2 , 6 , − 2] [ − 2 , 2 , 6] [3 , 1 , 2] [ − 1 , 3 , 4] [4 , − 1 , 3] [0 , 7 , − 1] 4 [6 , 2 , − 2] [ − 2 , 6 , 2] [ − 1 , 1 , 6] [3 , − 1 , 4] [4 , 3 , − 1] [ − 4 , 7 , 3] 17 [ − 5 , 2 , 9] [6 , − 2 , 2] [ − 1 , 6 , 1] [1 , − 1 , 6] [3 , 4 , − 1] [ − 4 , 3 , 7] [7 , − 4 , 3] 7 10 10 17 14 11 8 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 12 / 33

  12. A rational analogue Robin Sulzgruber On (rational) Shi tableaux March 2017 13 / 33

  13. A rational analogue Definition Let p be a positive integer relatively prime to the Coxeter number h . An alcove is called p -stable if its inverse lies inside the simplex bounded by the hyperplanes of height p . Robin Sulzgruber On (rational) Shi tableaux March 2017 13 / 33

  14. A rational analogue Definition Let p be a positive integer relatively prime to the Coxeter number h . An alcove is called p -stable if its inverse lies inside the simplex bounded by the hyperplanes of height p . Theorem (Thiel 2015) The number of p -stable alcoves equals p r . The number of dominant p -stable alcoves equals � r 1 ( p + e i ) . | W | i =1 Robin Sulzgruber On (rational) Shi tableaux March 2017 13 / 33

  15. Robin Sulzgruber On (rational) Shi tableaux March 2017 14 / 33

  16. 13 10 7 4 1 2 5 8 8 5 [5 , − 2 , 3] [0 , 5 , 1] [1 , 0 , 5] [2 , 4 , 0] [ − 3 , 2 , 7] 13 [ − 2 , 5 , 3] [0 , 1 , 5] [2 , 0 , 4] [4 , 2 , 0] 2 [ − 2 , 3 , 5] [2 , 1 , 3] [0 , 2 , 4] [4 , 0 , 2] 11 [2 , 3 , 1] [1 , 2 , 3] [0 , 4 , 2] 1 [3 , 2 , 1] [1 , 3 , 2] [ − 1 , 4 , 3] 10 [3 , 1 , 2] [ − 1 , 3 , 4] 4 [ − 1 , 1 , 6] [3 , − 1 , 4] 14 [1 , − 1 , 6] 7 [1 , 6 , − 1] 7 10 17 13 17 14 11 8 5 2 1 4 Robin Sulzgruber On (rational) Shi tableaux March 2017 14 / 33

  17. Robin Sulzgruber On (rational) Shi tableaux March 2017 15 / 33

  18. 16 13 10 7 4 1 2 5 11 16 8 8 [1 , 5 , 0] 5 [5 , − 2 , 3] [0 , 5 , 1] [1 , 0 , 5] [2 , 4 , 0] [ − 3 , 2 , 7] 13 [0 , 1 , 5] [2 , 0 , 4] 2 [ − 2 , 3 , 5] [2 , 1 , 3] [0 , 2 , 4] [4 , 0 , 2] 11 [2 , 3 , 1] [1 , 2 , 3] [0 , 4 , 2] 1 [3 , 2 , 1] [1 , 3 , 2] [ − 1 , 4 , 3] 10 [ − 2 , 2 , 6] [3 , 1 , 2] [ − 1 , 3 , 4] [4 , − 1 , 3] 4 [ − 1 , 1 , 6] [3 , − 1 , 4] 14 7 [1 , 6 , − 1] 7 10 17 13 17 14 11 8 5 2 1 4 Robin Sulzgruber On (rational) Shi tableaux March 2017 15 / 33

  19. Shi tableaux Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  20. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  21. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . The Shi tableau of w is the collection of the numbers t mh +1 ( α, w ) for α ∈ Φ + . Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  22. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . The Shi tableau of w is the collection of the numbers t mh +1 ( α, w ) for α ∈ Φ + . [2 , 0 , 4] [4 , 2 , 0] [0 , 2 , 4] [1 , 2 , 3] [0 , 4 , 2] Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  23. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . The Shi tableau of w is the collection of the numbers t mh +1 ( α, w ) for α ∈ Φ + . w = [4 , 2 , 0] [2 , 0 , 4] [4 , 2 , 0] [0 , 2 , 4] [1 , 2 , 3] [0 , 4 , 2] Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  24. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . The Shi tableau of w is the collection of the numbers t mh +1 ( α, w ) for α ∈ Φ + . w = [4 , 2 , 0] [2 , 0 , 4] [4 , 2 , 0] t 4 ( α 1 , w ) = 1 [0 , 2 , 4] [1 , 2 , 3] [0 , 4 , 2] Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  25. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . The Shi tableau of w is the collection of the numbers t mh +1 ( α, w ) for α ∈ Φ + . w = [4 , 2 , 0] [2 , 0 , 4] [4 , 2 , 0] t 4 ( α 1 , w ) = 1 [0 , 2 , 4] t 4 ( α 2 , w ) = 1 [1 , 2 , 3] [0 , 4 , 2] Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  26. Shi tableaux Definition (Fishel, Tzanaki, Vazirani 2011) Let w ( A ◦ ) be a dominant Shi alcove and α ∈ Φ + . Define t mh +1 ( α, w ) as the number of Shi hyperplanes of the form H α, k that separate w ( A ◦ ) and A ◦ . The Shi tableau of w is the collection of the numbers t mh +1 ( α, w ) for α ∈ Φ + . w = [4 , 2 , 0] [2 , 0 , 4] [4 , 2 , 0] t 4 ( α 1 , w ) = 1 [0 , 2 , 4] t 4 ( α 2 , w ) = 1 [1 , 2 , 3] [0 , 4 , 2] t 4 ( α 1 + α 2 , w ) = 1 Robin Sulzgruber On (rational) Shi tableaux March 2017 16 / 33

  27. Rational Shi tableaux Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  28. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  29. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . The rational Shi tableau of w is defined as the collection of numbers t p ( α, w ) for α ∈ Φ + . Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  30. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . The rational Shi tableau of w is defined as the collection of numbers t p ( α, w ) for α ∈ Φ + . 7 4 1 2 5 8 8 5 [2 , 4 , 0] [ − 3 , 2 , 7] 13 [2 , 0 , 4] 2 [0 , 2 , 4] [4 , 0 , 2] 11 [1 , 2 , 3] [0 , 4 , 2] 1 10 4 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  31. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . The rational Shi tableau of w is defined as the collection of numbers t p ( α, w ) for α ∈ Φ + . 7 4 1 2 5 8 8 w = [ − 3 , 2 , 7] 5 [2 , 4 , 0] [ − 3 , 2 , 7] 13 [2 , 0 , 4] 2 [0 , 2 , 4] [4 , 0 , 2] 11 [1 , 2 , 3] [0 , 4 , 2] 1 10 4 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  32. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . The rational Shi tableau of w is defined as the collection of numbers t p ( α, w ) for α ∈ Φ + . 7 4 1 2 5 8 8 w = [ − 3 , 2 , 7] 5 [2 , 4 , 0] [ − 3 , 2 , 7] 13 [2 , 0 , 4] 2 t 5 ( α 1 , w ) = 1 [0 , 2 , 4] [4 , 0 , 2] 11 [1 , 2 , 3] [0 , 4 , 2] 1 10 4 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  33. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . The rational Shi tableau of w is defined as the collection of numbers t p ( α, w ) for α ∈ Φ + . 7 4 1 2 5 8 8 w = [ − 3 , 2 , 7] 5 [2 , 4 , 0] [ − 3 , 2 , 7] 13 [2 , 0 , 4] 2 t 5 ( α 1 , w ) = 1 [0 , 2 , 4] [4 , 0 , 2] 11 t 5 ( α 2 , w ) = 1 [1 , 2 , 3] [0 , 4 , 2] 1 10 4 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  34. Rational Shi tableaux Definition Let w ( A ◦ ) be dominant and p -stable and α ∈ Φ + . Define t p ( α, w ) as the number of hyperplanes of the form H α, k with height less than p that separate w ( A ◦ ) and A ◦ . The rational Shi tableau of w is defined as the collection of numbers t p ( α, w ) for α ∈ Φ + . 7 4 1 2 5 8 8 w = [ − 3 , 2 , 7] 5 [2 , 4 , 0] [ − 3 , 2 , 7] 13 [2 , 0 , 4] 2 t 5 ( α 1 , w ) = 1 [0 , 2 , 4] [4 , 0 , 2] 11 t 5 ( α 2 , w ) = 1 [1 , 2 , 3] [0 , 4 , 2] 1 10 t 5 ( α 1 + α 2 , w ) = 2 4 5 2 1 4 7 Robin Sulzgruber On (rational) Shi tableaux March 2017 17 / 33

  35. The Main Conjecture Robin Sulzgruber On (rational) Shi tableaux March 2017 18 / 33

  36. The Main Conjecture Conjecture Every dominant p -stable element w ∈ � W is uniquely determined by its rational Shi tableau. Robin Sulzgruber On (rational) Shi tableaux March 2017 18 / 33

  37. The Main Conjecture Conjecture Every dominant p -stable element w ∈ � W is uniquely determined by its rational Shi tableau. Theorem The conjecture is true in type A n − 1 . Robin Sulzgruber On (rational) Shi tableaux March 2017 18 / 33

  38. The Main Conjecture Conjecture Every dominant p -stable element w ∈ � W is uniquely determined by its rational Shi tableau. Theorem The conjecture is true in type A n − 1 . Open Problem Characterise the set of rational Shi tableaux. Robin Sulzgruber On (rational) Shi tableaux March 2017 18 / 33

  39. Inverting the rational Shi tableau in type A n − 1 Robin Sulzgruber On (rational) Shi tableaux March 2017 19 / 33

  40. Inverting the rational Shi tableau in type A n − 1 Example Consider the affine permutation of type A 4 w = [7 , − 1 , 11 , 3 , − 5] . Robin Sulzgruber On (rational) Shi tableaux March 2017 19 / 33

  41. Inverting the rational Shi tableau in type A n − 1 Example Consider the affine permutation of type A 4 w = [7 , − 1 , 11 , 3 , − 5] . Then the alcove of w − 1 is contained in the simplex bounded by the hyperplanes of height p = 8. Robin Sulzgruber On (rational) Shi tableaux March 2017 19 / 33

  42. Inverting the rational Shi tableau in type A n − 1 Example Consider the affine The Shi tableau of w is given by permutation of type A 4 2 1 2 1 α 15 α 25 α 35 α 45 w = [7 , − 1 , 11 , 3 , − 5] . 1 2 0 α 14 α 24 α 34 2 1 Then the alcove of w − 1 is α 13 α 23 contained in the simplex bounded 0 α 12 by the hyperplanes of height p = 8. Robin Sulzgruber On (rational) Shi tableaux March 2017 19 / 33

  43. To Dyck paths via row-sums and column-sums 2 1 2 1 α 15 α 25 α 35 α 45 1 2 0 α 14 α 24 α 34 2 1 α 13 α 23 0 α 12 Robin Sulzgruber On (rational) Shi tableaux March 2017 20 / 33

  44. To Dyck paths via row-sums and column-sums 2 1 2 1 α 15 α 25 α 35 α 45 1 2 0 α 14 α 24 α 34 2 1 α 13 α 23 0 α 12 Robin Sulzgruber On (rational) Shi tableaux March 2017 20 / 33

  45. To long cycles (Ceballos, Denton, Hanusa 2016) Robin Sulzgruber On (rational) Shi tableaux March 2017 21 / 33

  46. To long cycles (Ceballos, Denton, Hanusa 2016) 12 13 11 8 9 10 13 7 12 11 6 10 3 4 5 9 2 7 8 6 1 5 4 1 2 3 Robin Sulzgruber On (rational) Shi tableaux March 2017 21 / 33

  47. To long cycles (Ceballos, Denton, Hanusa 2016) 12 13 11 8 9 10 13 7 12 11 6 10 3 4 5 9 2 7 8 6 1 5 4 1 2 3 (4 , 2 , 6 , 9 , 7 , 11 , 13 , 12 , 10 , 8 , 5 , 3 , 1) Robin Sulzgruber On (rational) Shi tableaux March 2017 21 / 33

  48. Back to Dyck paths (Ceballos, Denton, Hanusa 2016) Robin Sulzgruber On (rational) Shi tableaux March 2017 22 / 33

  49. Back to Dyck paths (Ceballos, Denton, Hanusa 2016) (4 , 2 , 6 , 9 , 7 , 11 , 13 , 12 , 10 , 8 , 5 , 3 , 1) Robin Sulzgruber On (rational) Shi tableaux March 2017 22 / 33

  50. Back to Dyck paths (Ceballos, Denton, Hanusa 2016) (4 , 2 , 6 , 9 , 7 , 11 , 13 , 12 , 10 , 8 , 5 , 3 , 1) Robin Sulzgruber On (rational) Shi tableaux March 2017 22 / 33

  51. Back to Dyck paths (Ceballos, Denton, Hanusa 2016) (4 , 2 , 6 , 9 , 7 , 11 , 13 , 12 , 10 , 8 , 5 , 3 , 1) Robin Sulzgruber On (rational) Shi tableaux March 2017 22 / 33

  52. To n and p flush abaci (Anderson 2002) Robin Sulzgruber On (rational) Shi tableaux March 2017 23 / 33

  53. To n and p flush abaci (Anderson 2002) Robin Sulzgruber On (rational) Shi tableaux March 2017 23 / 33

  54. To n and p flush abaci (Anderson 2002) 40 35 30 25 20 15 10 5 0 32 27 22 17 12 7 2 -3 -8 24 19 14 9 4 -1 -6 -11 -16 16 11 6 1 -4 -9 -14 -19 -24 8 3 -2 -7 -12 -17 -22 -27 -32 0 -5 -10 -15 -20 -25 -30 -35 -40 Robin Sulzgruber On (rational) Shi tableaux March 2017 23 / 33

  55. To n and p flush abaci (Anderson 2002) 40 35 30 25 20 15 10 5 0 32 27 22 17 12 7 2 -3 -8 24 19 14 9 4 -1 -6 -11 -16 16 11 6 1 -4 -9 -14 -19 -24 8 3 -2 -7 -12 -17 -22 -27 -32 0 -5 -10 -15 -20 -25 -30 -35 -40 Robin Sulzgruber On (rational) Shi tableaux March 2017 23 / 33

  56. To n and p flush abaci (Anderson 2002) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -14 -13 -12 -11 -10 40 35 30 25 20 15 10 5 0 -9 -8 -7 -6 -5 32 27 22 17 12 7 2 -3 -8 -4 -3 -2 -1 0 24 19 14 9 4 -1 -6 -11 -16 1 2 4 3 5 16 11 6 1 -4 -9 -14 -19 -24 6 7 9 8 10 8 3 -2 -7 -12 -17 -22 -27 -32 12 11 13 14 15 0 -5 -10 -15 -20 -25 -30 -35 -40 17 16 18 19 20 21 22 23 24 25 . . . . . . . . . . . . . . . Robin Sulzgruber On (rational) Shi tableaux March 2017 23 / 33

  57. Shift back to affine permutations (Lascoux 2001) Robin Sulzgruber On (rational) Shi tableaux March 2017 24 / 33

  58. Shift back to affine permutations (Lascoux 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 4 3 5 6 7 9 8 10 12 11 13 14 15 17 16 18 19 20 21 22 23 24 25 . . . . . . . . . . . . . . . Robin Sulzgruber On (rational) Shi tableaux March 2017 24 / 33

  59. Shift back to affine permutations (Lascoux 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -14 -13 -12 -11 -10 -19 -18 -17 -16 -15 -9 -8 -7 -6 -5 -14 -13 -12 -11 -10 -4 -3 -2 -1 -9 -8 -6 -5 0 -7 1 2 4 -3 -1 0 3 5 -4 -2 6 7 9 2 5 8 10 1 3 4 12 10 11 13 14 15 6 7 8 9 17 16 18 19 20 11 12 13 14 15 21 22 23 24 25 16 17 18 19 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robin Sulzgruber On (rational) Shi tableaux March 2017 24 / 33

  60. Shift back to affine permutations (Lascoux 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -14 -13 -12 -11 -10 -19 -18 -17 -16 -15 -9 -8 -7 -6 -5 -14 -13 -12 -11 -10 -4 -3 -2 -1 -9 -8 -6 -5 0 -7 w − 1 = [ − 7 , − 4 , 4 , 7 , 15] 1 2 4 -3 -1 0 3 5 -4 -2 6 7 9 2 5 8 10 1 3 4 12 10 11 13 14 15 6 7 8 9 17 16 18 19 20 11 12 13 14 15 21 22 23 24 25 16 17 18 19 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robin Sulzgruber On (rational) Shi tableaux March 2017 24 / 33

  61. Shift back to affine permutations (Lascoux 2001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -14 -13 -12 -11 -10 -19 -18 -17 -16 -15 -9 -8 -7 -6 -5 -14 -13 -12 -11 -10 -4 -3 -2 -1 -9 -8 -6 -5 0 -7 w − 1 = [ − 7 , − 4 , 4 , 7 , 15] 1 2 4 -3 -1 0 3 5 -4 -2 w = [7 , − 1 , 11 , 3 , − 5] 6 7 9 2 5 8 10 1 3 4 12 10 11 13 14 15 6 7 8 9 17 16 18 19 20 11 12 13 14 15 21 22 23 24 25 16 17 18 19 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robin Sulzgruber On (rational) Shi tableaux March 2017 24 / 33

  62. This is the end. Thank you! Robin Sulzgruber On (rational) Shi tableaux March 2017 25 / 33

  63. nl. Hl. h;3 * l;, r,ii," * [l-, 11 H,r,-, .r/ 4/ Shi coordinates Robin Sulzgruber On (rational) Shi tableaux March 2017 26 / 33

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