|
| 1 | +.. -*- coding: utf-8 -*- |
| 2 | + |
| 3 | +.. _lascoux.factorization_gem: |
| 4 | + |
| 5 | +============================================================================================================ |
| 6 | +Alain Lascoux finding a gem by computer exploration: Factorization properties of Young's natural idempotents |
| 7 | +============================================================================================================ |
| 8 | + |
| 9 | +About this worksheet |
| 10 | +-------------------- |
| 11 | + |
| 12 | +This worksheet was produced at the occasion of a tribute to Alain |
| 13 | +Lascoux at `FPSAC'14 <http://fpsac.org/confs/fpsac-2014/`_. |
| 14 | + |
| 15 | +A computer exploration |
| 16 | +---------------------- |
| 17 | + |
| 18 | +Let's build one of Young's natural idempotents for the Symmetric group |
| 19 | +:math:`S_9`:: |
| 20 | + |
| 21 | + sage: W = SymmetricGroup(9) |
| 22 | + sage: I = Partition([3,3,2,1]) |
| 23 | + sage: J = Composition([2,3,4]) |
| 24 | + sage: tI = StandardTableaux(I).last() |
| 25 | + sage: tJ = Tableau([[6,7,8,9], [3,4,5], [1,2]]) |
| 26 | + sage: tIc = Tableau([[3,6,8,9],[2,5,7],[1,4]]) |
| 27 | + sage: muI = W([1,4,2,5,7,3,6,8,9]) |
| 28 | + |
| 29 | +It's indexed by a pair of standard tableaux, which we show here, in |
| 30 | +French notation of course:: |
| 31 | + |
| 32 | + sage: Tableaux.global_options(convention="French") |
| 33 | + sage: tI.pp() |
| 34 | + sage: print |
| 35 | + sage: tJ.pp() |
| 36 | + 9 |
| 37 | + 7 8 |
| 38 | + 4 5 6 |
| 39 | + 1 2 3 |
| 40 | + |
| 41 | + 1 2 |
| 42 | + 3 4 5 |
| 43 | + 6 7 8 9 |
| 44 | + |
| 45 | +The idempotent is the usual product of two pieces, a sum across a row |
| 46 | +stabilizer, and an alternating sum across a column stabilizer:: |
| 47 | + |
| 48 | + sage: A = W.algebra(QQ) |
| 49 | + sage: squareI = A.sum_of_monomials(W(sigma) |
| 50 | + ....: for sigma in tI.row_stabilizer()) |
| 51 | + sage: nablaJ = A.sum_of_terms ([W(sigma), sigma.sign()] |
| 52 | + ....: for sigma in tJ.row_stabilizer()) |
| 53 | + ....: |
| 54 | + sage: squareI |
| 55 | + <html>...</html> |
| 56 | + |
| 57 | +Both pieces being large, their product is a huge linear combination of |
| 58 | +permutations. One can compute with it, but it's useless to even look |
| 59 | +at it:: |
| 60 | + |
| 61 | + sage: idempotent = nablaJ * A.monomial(muI) * squareI |
| 62 | + sage: len(idempotent) |
| 63 | + <html>...</html> |
| 64 | + |
| 65 | +So Alain went onto a quest for a compact representation of this object |
| 66 | +that would be amenable to scrutiny and hand manipulation. |
| 67 | + |
| 68 | +The first step, quite natural, was to represent permutations by their |
| 69 | +Lehmer code. The second step, typical of Alain, was to encode each |
| 70 | +such code as an exponent vector. This makes the idempotent into a huge |
| 71 | +multivariate polynomial:: |
| 72 | + |
| 73 | + sage: P = QQ["x1,x1,x2,x3,x4,x5,x6,x7,x8,x9"] |
| 74 | + sage: x = muI(P.gens()) |
| 75 | + sage: def to_monomial(sigma): |
| 76 | + ....: code = Permutation(sigma).to_lehmer_code() |
| 77 | + ....: return prod( xi^ci for xi,ci in zip(x,code) ) |
| 78 | + sage: to_polynomial = A.module_morphism(to_monomial, codomain=P) |
| 79 | + sage: p = to_polynomial(idempotent) |
| 80 | + |
| 81 | +Here are its first 20 terms:: |
| 82 | + |
| 83 | + sage: sum(p.monomials()[:20]) |
| 84 | + <html>...</html> |
| 85 | + |
| 86 | +So far, so good. But the gain is not that obvious. |
| 87 | + |
| 88 | +Now comes the step of genius, because it is so unnatural: the |
| 89 | +multiplicative structure of the algebra of the symmetric group has |
| 90 | +nothing to do with that of polynomials. There is no reason whatsoever |
| 91 | +to believe that the multiplication of polynomials would have any |
| 92 | +**meaning**. |
| 93 | + |
| 94 | +Yet, Alain tried to actually factor that polynomial, and here is the |
| 95 | +gem that came out:: |
| 96 | + |
| 97 | + sage: factor(p) |
| 98 | + <html>...</html> |
| 99 | + |
| 100 | +Reference |
| 101 | +--------- |
| 102 | + |
| 103 | +- *Young's natural idempotents as polynomials* , Alain Lascoux, Annals of Combinatorics 1, 1997, 91-98 |
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