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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<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!
<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
<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)