Please have a look at this presentation on Young tableaux,
I'm trying to understand the signs I mention there - what do the $\otimes$ and $\oplus$ symbols mean?
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2Direct product and direct sum – Slereah Apr 18 '18 at 15:42
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If you want to typeset a symbol in MathJax but you don't know the command, use Detexify to identify it. – Emilio Pisanty Apr 18 '18 at 16:46
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Related: What does "the ${\bf N}$ of a group" mean?. – Emilio Pisanty Apr 18 '18 at 16:50
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Related $SU(3)$ post: http://physics.stackexchange.com/q/10403/2451 and links therein; especially the answer http://physics.stackexchange.com/a/14586/2451 . – Qmechanic Apr 18 '18 at 17:36
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The symbols $\oplus$ and $\otimes$ are generally used to denote direct sums and direct products, with the precise details depending on the context (i.e. depending on whether you're talking about groups or vector spaces or some other algebraic structure).
In the group-representation context embodied by these Young tableaux, the $\oplus$ symbol denotes the direct sum of the vector spaces of two representations, and the $\otimes$ symbol denotes their tensor product, with both implicitly conferring a group representation to the resulting vector space (i.e. a mapping $g:G\to U\oplus V$ or $g:G\to U\otimes V$) given by the direct sum and tensor product, respectively, of the individual factors' representations.
Emilio Pisanty
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