Suffice to say I used them a lot and had tons of trouble keeping the computations straight.
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Replying to @St_Rev @alicemazzy
Even the scent of numbers made me nauseous as a student. Spozally-abstract algebra that’s covertly about polynomials made me want to run. I’ve come to appreciate continuity in my golden years
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Replying to @Meaningness @alicemazzy
What's really going on is: Take a vector space V and let sigma be the flip operator on V \tensor V: sigma(x \tensor y) = y \tensor x Then we call the +1 eigenspace of sigma the symmetric product and the -1 eigenspace the external product.
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Replying to @St_Rev @alicemazzy
*consults wiki, having gotten lost at “eigenspace”*
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I must have known this once: “The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T.”
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So by -1 eigenspace, we mean the elements that just go backward after transformation? And +1 are those unaffected?
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Replying to @Meaningness @alicemazzy
Sorry, yeah, the symmetric space is spanned by elements of the form (x \tensor x) and (x \tensor y + y \tensor x), exterior by (x \tensor y - y \tensor x). You can check that the flip operator preserves all the former and multiplies the latter by -1
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Replying to @St_Rev @alicemazzy
Is there some sort of intuition for why I should care about this? (Sorry to be a dumb computer scientist)
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Replying to @Meaningness @alicemazzy
It lets you automagically decompose a space/algebra/module of dimension n^2 into two pieces of dimension (n^2 + n)/2 and (n^2 - n)/2. Assume you have an algorithm that scales as dim^4 or 2^dim...
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It also lets you decompose higher-order powers V^(\tensor 2n) into correspondingly more pieces. Also those pieces tend to have interesting properties, eg https://en.wikipedia.org/wiki/Symplectic_geometry … uses exterior powers on the cotangent space of a manifold to do something or other
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OK, I was looking at that same Wiki article a couple of months ago for an unrelated reason (coming at it from Hamiltonian mechanics), so maybe someday I need to understand this!
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