Kronecker product recovering the initial representation?

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SUMMARY

The discussion focuses on the properties of the Kronecker product in relation to representation theory, specifically examining the decomposition of products like 27 x 27 in E6 and 3 x 3 in SU(3). It highlights that the product can yield the original representation or its conjugate within the resulting sum. The conversation seeks to establish general lemmas that dictate when an N x N product recovers the original representation or its conjugate, noting that R x R consistently decomposes into symmetric and alternating squares.

PREREQUISITES
  • Understanding of representation theory in mathematics
  • Familiarity with the Kronecker product
  • Knowledge of groups such as E6 and SU(3)
  • Concepts of symmetric and alternating representations
NEXT STEPS
  • Research the properties of the Kronecker product in representation theory
  • Study the decomposition of representations in E6 and SU(3)
  • Explore lemmas related to representation recovery in group theory
  • Investigate symmetric and alternating squares in linear algebra
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Mathematicians, theoretical physicists, and students studying representation theory or group theory, particularly those interested in the properties of the Kronecker product and its applications in various mathematical contexts.

arivero
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In E6, the product 27 x 27 contains the (conjugate) 27. In SU(3), something similar happens with 3 x 3, which decomposes as 3 + 6.

I was wondering, how usual is this? Do we have some lemmas telling when a product N x N is going to "recover" the original N, or its conjugate, inside the sum?
 
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so, what happens is that R x R always decompose in a sum of the symmetric plus alternating (or exterior) square.
 

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