#15650: Permutations and symmetric group algebra: getting rid of 'mult' global
in
seminormal form
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Reporter: darij | Owner:
Type: defect | Status: needs_review
Priority: major | Milestone: sage-6.2
Component: combinatorics | Resolution:
Keywords: permutation, | Merged in:
sage-combinat, descent algebra, | Reviewers: Travis Scrimshaw
symmetric group, seminormal form | Work issues:
Authors: Darij Grinberg | Commit:
Report Upstream: N/A | 473853757fce07a9b6ee497f2821e7161ed8989c
Branch: | Stopgaps:
public/combinat/seminormal |
Dependencies: |
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Comment (by andrew.mathas):
Hi Darij,
To work with the seminormal form you don't need to compute the idempotents
at all. Rather you just need a basis `{ v_t }` together with an
implementation of the action
{{{
s_r v_t = 1/\rho_r(t)v_t + (1+\rho_r(t))/\rho_r(t) v_{s_r t},
}}}
where `\pho_r(t)=c_{r+1}(t)-c_r(t)` is the axial distance from `r` to
`r+1` inside the tableau `t`. Here I use the convention that `v_{s_r t}=0`
if `s_rt` is not standard.
Of course, there are many other equivalent ways to write down the
seminormal form, but this is my favourite version because it is very
symmetric and it is defined over the rationals.
The action of the idempotents on this basis of the semisimple Specht is
easy:
{{{F_s v_t=\delta_[st} v_t}}}
This follows because `L_r v_t = c_r(t)v_t`, for all `r`.
It is only when you need to work with the idempotents `F_t`, given
explicitly as elements of the group algebra, that you ever need to use
these formulas.
Andrew
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Ticket URL: <http://trac.sagemath.org/ticket/15650#comment:25>
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