#18798: Jucys-Murphy Elements for Brauer Algebra
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Reporter: ghseeli | Owner:
Type: enhancement | Status: needs_review
Priority: minor | Milestone: sage-6.10
Component: algebra | Resolution:
Keywords: days65, partition | Merged in:
algebra, diagram algebra, jucys- | Reviewers:
murphy | Work issues:
Authors: George H. | Commit:
Seelinger | 15aa5a132c33abfa833a6f5a8974bcf93b3776b0
Report Upstream: N/A | Stopgaps:
Branch: |
u/ghseeli/jucys_murphy_elements_for_brauer_algebra|
Dependencies: #18762 |
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Comment (by tscrim):
A couple of suggestions:
{{{#!diff
def jucys_murphy(self, j):
r"""
- Return a generalized Jucys-Murphy elements for the Brauer
algebra.
- These are outlined in [Naz]_.
+ Return the ``j``-th generalized Jucys-Murphy element of ``self``.
+ Could you give a more detailed description of how the elements
are defined?
+
REFERENCES:
.. [Naz] Maxim Nazarov, Young's Orthogonal Form for Brauer's
Centralizer
- Algebra. Journal of Algebra 182 (1996), 664--693.
+ Algebra. Journal of Algebra 182 (1996), 664--693.
EXAMPLES:
sage: z = var('z')
sage: B = BrauerAlgebra(3,z)
sage: B.jucys_murphy(1)
1/2*z - 1/2
sage: B.jucys_murphy(3)
- -B{{-3, -2}, {-1, 1}, {2, 3}} - B{{-3, -1}, {-2, 2}, {1, 3}}
+ B{{-3, 1}, {-2, 2}, {-1, 3}} + B{{-3, 2}, {-2, 3}, {-1, 1}} +
(1/2*z-1/2)*B{{-3, 3}, {-2, 2}, {-1, 1}}
+ -B{{-3, -2}, {-1, 1}, {2, 3}} - B{{-3, -1}, {-2, 2}, {1, 3}}
+ + B{{-3, 1}, {-2, 2}, {-1, 3}} + B{{-3, 2}, {-2, 3}, {-1, 1}}
+ + (1/2*z-1/2)*B{{-3, 3}, {-2, 2}, {-1, 1}}
"""
- B = self
- return (B._q-1)/2 + sum(B([[i,-j],[j,-i]]) - B([[i,j],[-i,-j]])
for i in range(1,j))
+ I = self._indices
+ one = self.base_ring().one()
+ return ((self._q-1)/2
+ + self._from_dict({I([[i,-j],[j,-i]]): one for i in
range(1,j)}, remove_zeros=False)
+ - self._from_dict({I([[i,j],[-i,-j]]): one for i in
range(1,j)}, remove_zeros=False))
}}}
Do you think we should also make this a cached method? What about when `j
> n`, what happens then (or is it even defined)?
--
Ticket URL: <http://trac.sagemath.org/ticket/18798#comment:11>
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