UmbralRaptor changed the topic of #kspacademia to: https://gist.github.com/pdn4kd/164b9b85435d87afbec0c3a7e69d3e6d | Dogs are cats. Spiders are cat interferometers. | Космизм сегодня! | Document well, for tomorrow you may get mauled by a ネコバス. | <UmbralRaptor> egg|nomz|egg: generally if your eyes are dewing over, that's not the weather. | <ferram4> I shall beat my problems to death with an engineer. | We can haz pdf
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<SilverFox> whitequark, do you know why powerbanks dont do pass-through charging? like they cant charge and discharge at the same time?
<UmbralRaptop> SnoopJeDi: belatedly: ¯\_(ツ)_/¯
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<mlbaker> egg|cell|egg: what is {–,–} here?
<galois> title: [1912.04903] Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology
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<mlbaker> why did it respond with ¯\_(ツ)_/¯ before
<egg|laptop|egg> mlbaker: the anticommutator
<mlbaker> oh
<egg|laptop|egg> (of endomorphisms of V)
<egg|laptop|egg> (where V∧V = 𝔰𝔬(V) ⊆ End(V))
<egg|laptop|egg> mlbaker: it's interesting that {I, _} is tr I in 2d, and I∧I the determinant; I wonder whether there's a way to generalize the former to multivectors beyond bivectors to also get the trace on pseudoscalars
<egg|laptop|egg> (in higher dimensions that is)
<mlbaker> i'm pretty sure this is just some familiar operation in disguise
<mlbaker> but it's eluding me at the moment
<mlbaker> some kind of tensor contraction
<egg|laptop|egg> mlbaker: re. ¯\_(ツ)_/¯, I think SnoopJeDi fixed a bug in galois
<egg|laptop|egg> hm. so {I, _} maps v∧w to Iv∧w + v∧Iw (used the symmetry of I here)
<egg|laptop|egg> so that points towards the generalization; but what *is* this operation...
<egg|laptop|egg> I think in index notation that's (I^k_i + I^k_j)?
<egg|laptop|egg> hm that doesn't make sense
<egg|laptop|egg> yeah no ignore my index notation nonsense
<mlbaker> i asked a friend whose math power level is way above mine and he said he'd have to think about it more and "maybe it's some sort of super lie algebra representation"
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<whitequark> SilverFox: "it's more effort than not doing it"
<whitequark> there's no reason they can't, they just choose not to
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<egg|work|egg> mapping v∧w to Iv∧w + v∧Iw is puzzling, it doesn't seem to require I to be symmetric? whereas {I, _} only works if I is symmetric
<egg|work|egg> (Does it actually map v∧w to Iv∧w + v∧Iw?)
<SnoopJeDi> mlbaker, egg|work|egg the shrug reaction was because apparently lxml.html does not support documents in a unicode (str, in Py3) object with an XML encoding declared. I wrote a little check that re-encodes before passing if needs be, which is probably fragile, but it works!
<egg|work|egg> http://www.tcheb.ru/
<galois> title: Механизмы П. Л. Чебышева
<egg|work|egg> (just testing a non-ASCII title)
<SnoopJeDi> yea it should handle those fine, the problem before was an exception being raised when parsing the document
<UmbralRaptop> egg: chonky cat
<egg|cell|egg> Chonk
<SnoopJeDi> !how chonk
<galois> SnoopJeDi: 975 chonks
<SnoopJeDi> o lawd
<SnoopJeDi> !wpn -add:adj chonky
<galois> Added adj 'chonky'
<SnoopJeDi> !wpn -add:wpn chonk
<galois> Added wpn 'chonk'
<UmbralRaptop> !wpn SnoopJeDi
* galois gives SnoopJeDi a hypothetical distributed curse
<SnoopJeDi> that's very laundry files of you, galois
<SnoopJeDi> !wpn UmbralRaptop
* galois gives UmbralRaptop a modular moon 猫
<UmbralRaptop> Ah, a cryptocurrency whitepaper
<SnoopJeDi> crypomagick
<UmbralRaptop> mooncat
* UmbralRaptop 🔪 🚌
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<egg|cell|egg> Mlbaker: v∧w ↦ Iv∧w + v∧Iw is B↦IB+BᵗI, so that's the nonsymmetric generalisation
<egg|cell|egg> It's the anticommutator only for symmetric I
<UmbralRaptop> ahh, wedge products
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<egg> mlbaker: so I guess the question now becomes "WTF is v∧w ↦ Iv∧w + v∧Iw"
<egg> (and its generalizations to higher multivectors)
<mlbaker> u∧v∧...∧w ↦ Iu∧v∧...w + u∧Iv∧...∧w + ... + u∧v∧...∧Iw?
<egg> mlbaker: yeah
<mlbaker> reminds me of the tensor product of two lie algebra representations
<mlbaker> except there are wedges
<egg> mlbaker: when u∧v∧...∧w is a pseudoscalar, that's the trace btw
<mlbaker> right
<egg> so it is to Λ^n I what the trace is to the determinant
<egg> not sure what that tells us :-p
<mlbaker> hmmmm
<mlbaker> where's richard p. stanley when you need him
<galois> title: Schur functor in nLab
<mlbaker> "Mathematicians often work with a decategorified version of Schur: its Grothendieck group, also known as the ring of symmetric functions."
<mlbaker> hmmmm
<mlbaker> i think there must be something going on here
<egg|laptop|egg> mlbaker: wait how is it related to enumerative combinatorics
<mlbaker> well we know /\^n is det, and your thing is kinda similar but it's giving the trace
<mlbaker> vol 2 of enumerative combinatorics is all about symmetric functions
<egg|laptop|egg> ah
<mlbaker> the thing though is your thing doesn't appear to be functorial
<mlbaker> im also not entirely convinced it's well-defined?
<egg|laptop|egg> mlbaker: how so
<mlbaker> v tensor w maps to Iv∧w + v∧Iw... so v tensor w - w tensor v maps to Iv∧w + v∧Iw - Iw∧v - w∧Iv = Iv∧w + v∧Iw + v∧Iw + Iv∧w = 2(Iv∧w + v∧Iw)
<mlbaker> and... why is this zero?
<mlbaker> (am trying to check that this map ∧^2 -> ∧^2 does indeed exist by first defining a bilinear map from V tensor V, checking it's antisymmetric, and invoking universal property of the exterior power)
<mlbaker> it does seem to be alternating (v∧v |-> Iv∧v + v∧Iv = Iv∧v - Iv∧v = 0), so I'm assuming I made a silly mistake somewhere
<mlbaker> er, that v∧v should've been v tensor v
<egg|laptop|egg> I'm mildly confused as to why you're starting from tensor products here
<egg|laptop|egg> but v∧w *plus* w∧v does map to 0
<mlbaker> OH
<egg|laptop|egg> !wpn -add:wpn sign
<galois> Added wpn 'sign'
<mlbaker> right that's what i need to quotient by lol
<egg|laptop|egg> symmetric, antisymmetric, same thing :-p
<egg|laptop|egg> the wise man bowed his head solemnly and spoke: "theres actually zero difference between symmetric & antisymmetric maps" etc.
<mlbaker> umm
<egg|laptop|egg> "invoking universal property of the exterior power" sounds far more magical than it is
<mlbaker> okay
<mlbaker> so to what extent can we get away with this sort of thing
<SilverFox> whitequark, how hard would it be to implement into a cracked open battery bank?
<egg|laptop|egg> mlbaker: what do you mean
<mlbaker> like, can we do one on every ∧^k where we just map v1∧...∧vk to \sum_{i<j} v1∧...∧Ivi∧...∧Ivj∧...∧vk, say?
<egg|laptop|egg> hmm
<egg|laptop|egg> mlbaker: I think you would have to sum over all subsets of cardinality n of {1 .. k}
<egg|laptop|egg> in order for the quotient set to remain happy
<egg|laptop|egg> in particular when n = k you would get your determinanty functor back?
<mlbaker> this doesn't seem to be obviously alternating
<egg|laptop|egg> hm
<mlbaker> suppose v1=v2, then if i=1 and j=2 the summand is dead, and if i,j are both larger than 2 the summand is dead
<mlbaker> but if i=1 and j=3 say...
<egg|laptop|egg> there's something disturbing to doing that sum over the i<j, you rely on the ordering of the basis which feels evil
<egg|laptop|egg> wait no nevermind
<egg|laptop|egg> hm
<mlbaker> oh wait
<mlbaker> the i=1, j=3 term would cancel with the i=2, j=3 one...
<mlbaker> maybe this actually does work? o_O
<egg|laptop|egg> what about the subset thing?
<mlbaker> isn't that what i did?
<mlbaker> sum is over all cardinality 2 subsets
<egg|laptop|egg> oh right
<egg|laptop|egg> I misread that as applying I to v_i through v_j
<egg|laptop|egg> which seemed very fishy
<mlbaker> ah
<egg|laptop|egg> yeah I think it works summing over all cardinality n subsets
<mlbaker> this probably comes down to
<mlbaker> setting x=-1 in (1+x) = \sum_k \binom{n}{k} x^k
<mlbaker> lol
<egg|laptop|egg> ... which means that if you do that on the pseudoscalars, you have a sequence of things that go from the trace to the determinant; what would those be?
<egg|laptop|egg> the treterminant
<mlbaker> well
<mlbaker> they would be tr( /\^k T ), of course
<mlbaker> up to sign the coefficients of the char poly of T
<mlbaker> it's a bit strange though because your thing is still acting on the top exterior power, yet gives multiplication by the trace
<mlbaker> actually this is a thing
<mlbaker> do you remember the proof of why the lie algebra of SL is the traceless things
<mlbaker> you take a curve gamma(t) in SL with gamma(0)=I, and let X := gamma'(0)
<mlbaker> then you note gamma(t)e_1 ∧ ... ∧ gamma(t)e_n = e_1 ∧ ... ∧ e_n for all t
<mlbaker> differentiate at t=0 and you get \sum_i e_1 ∧ ... ∧ Xe_i ∧ ... ∧ e_n = tr(X) e_1 ∧ ... ∧ e_n = 0
<egg|laptop|egg> I do not remember this proof (though I may have seen it), but yeah, that's our map
<UmbralRaptop> Ablate is real https://arxiv.org/abs/2001.02217
<galois> title: [2001.02217] An ultra-short period rocky super-Earth orbiting the G2-star HD 80653
<egg|laptop|egg> mlbaker: so, do any of those things have a name :-p
<mlbaker> why am i having so much trouble articulating this
<mlbaker> the point is that like
<mlbaker> let G be a Lie group
<mlbaker> if you have a representation rho : G -> GL(V)
<mlbaker> then you get an induced rep G -> GL(/\^n V) by letting /\^n(rho)(g) := /\^n[ rho(g) ]
<mlbaker> now
<mlbaker> the infinitesimal representation g -> gl(/\^n V) is
<mlbaker> d[/\^n(rho)](X) : v_1 ∧ ... ∧ v_n |-> \sum_i v_1 ∧ ... ∧ Xv_i ∧ ... ∧ v_n
<mlbaker> so of course it's not a functor, because composition is not what matters, the lie bracket is
<egg|laptop|egg> hm, right
<egg|laptop|egg> (in my application X would be symmetric rather than antisymmetric so it doesn't really live naturally in a Lie algebra but that's a different question)
<egg|laptop|egg> mlbaker: and what about the higher maps, that do things to a subset of the v_i on each term?
<egg|laptop|egg> !u ⋀
<galois> ⋀: U+22c0 N-ARY LOGICAL AND
<mlbaker> hmm oops
<mlbaker> that Xv_i should be uh
<mlbaker> drho(X)v_i
<mlbaker> this is weird
<egg|laptop|egg> :D
<mlbaker> okay, so the statement that /\^k is a functor says that for every pair of v.s. V and W, you get an induced map Hom(V,W) -> Hom(/\^k V, /\^k W)
<egg|laptop|egg> yeah our thing is really about endomorphisms
<mlbaker> if we put V=W, this goes End(V) -> End(/\^k V), and it preserves the group of units
<egg|laptop|egg> the group of units?
<mlbaker> ie GL(V) -> GL(/\^k V)
<egg|laptop|egg> ah
<mlbaker> the map End(V) -> End(/\^k V) is not an algebra morphism
<mlbaker> because addition is doing god knows what
<mlbaker> but it is multiplicative
<egg|laptop|egg> yeah
<mlbaker> now if G and H are Lie groups and you have a group homomorphism G->H
<egg|laptop|egg> and I think the first map here (applying it to one of the factors in each term) is additive?
<mlbaker> taking pushforward/differential at the identity yields a Lie algebra morphism g->h
<mlbaker> and that's where your map comes from
<egg|laptop|egg> right.
<egg|laptop|egg> and the intermediate maps, with subsets of the factors?
<egg|laptop|egg> :D
<mlbaker> so somehow this thing is very closely related to the exterior power functors
<mlbaker> as for the ones where you apply it on more than one factor
<mlbaker> i'm not sure if those arise this way
<mlbaker> ?
<mlbaker> also i should be careful here
<mlbaker> my k should be dim(V)
<mlbaker> in order to get your map
<egg|laptop|egg> hm but then that only is the map that gives the trace, not the map on V∧V?
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<mlbaker> for k < dim(V) idk what the differential is
<mlbaker> ummm
<mlbaker> the map on V∧V sends e_i∧e_j to Te_i∧e_j + e_i∧Te_j = (\sum_k T_{ki}e_k)∧e_j + e_i∧(\sum_k T_{kj}e_k)
<mlbaker> which, like, what the heck is that
<egg|laptop|egg> :D
<mlbaker> i think it's still the differential though
<mlbaker> of the map /\^2(-) : GL(V) -> GL(/\^2 V)