#8442: Lie Methods and Related Combinatorics (tutorial)
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Reporter: bump | Owner: bump
Type: enhancement | Status: needs_info
Priority: major | Milestone: sage-4.6.1
Component: documentation | Keywords:
Author: Daniel Bump | Upstream: N/A
Reviewer: Minh Van Nguyen, Mark Jordan | Merged:
Work_issues: |
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Comment(by bump):
> Basically, that meant computing certain symmetric power of certain
representation of GL_m (or SL_m), and finding out whether there was a
1-dimensional sub-representation.
For symmetric (or exterior) square you can use the
frobenius_schur_indicator method. For higher symmetric powers, here is a
way. Suppose we want to compute the symmetric 5-th power of the
8-dimensional adjoint representation of SL(3). The relevant groups are
SL(3) and SL(8), so we need Weyl Character rings A2 and A7.
{{{
sage: A2=WeylCharacterRing("A2",style="coroots")
sage: A7=WeylCharacterRing("A7",style="coroots",cache="true")
sage: s = A7.fundamental_weights()[1]
sage: A7(5*s)
A7(5,0,0,0,0,0,0)
sage: A7(5*s)
A7(5,0,0,0,0,0,0)
sage: A7(5*s).degree()
792
}}}
This is the symmetric 5-th power of the standard representation of SL(8),
which we want to branch down to SL(3) along the adjoint representation,
which is a homomorphism SL(3) --> SL(8). So we create the adjoint
representation, then branch the symmetric 5-th power representation of
SL(8) down to SL(3).
{{{
sage: ad=A2(1,1); ad.degree()
8
sage: A7(5*s).branch(A2,rule=branching_rule_from_plethysm(ad,"A7"))
A2(0,0) + 2*A2(1,1) + A2(0,3) + A2(3,0) + 2*A2(2,2) + A2(1,4) + A2(4,1) +
2*A2(3,3) + A2(2,5) + A2(5,2) + A2(4,4) + A2(5,5)
}}}
There is your decomposition into irreducibles. You can see that there is a
copy of the trivial representation.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8442#comment:48>
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