#Zambroni

[underachievers-please-try-harder]

Track of the day // Wicketkeeper - Alarm Clock Radio

[goofingup]

my bet’s on no. [x]

[earhartsease]

my little pony on ice so fans can call themselves zambronies

[noloveforned]

tune into wlur from 8pm until midnight tonight for four hours of no love for ned. i'm heading down to north carolina this weekend to meet up with friends and catch a couple yo la tengo shows but i've got a new show all queued up.

if you're not the "schedule my life" type, you can also check out last week's show on mixcloud whenever you'd like... and if you click around a little bit, you'll find tonight's there too!

no love for ned on wlur �� february 23rd, 2024 from 8-10pm

artist // track // album // label car seat headrest // cute thing // twin fantasy // matador apollo ghosts // no one knows your mind // amethyst ep // you've changed alien nosejob // west side story // cold bare facts 7" // anti fade busted head racket // sad little heart // junk food // painscale pouty // now // take me to honey island cassette // bratty office dog // big air // spiel // new west hooper crescent // wrong direction // essential tremors // spoilsport phil and the tiles // dry run // double happiness // legless rosie tucker // all my exes live in vortexes // utopia now! // sentimental puritan // the captives // how to move a piano // elsinor eggy // open field // open field 7" // (self-released) can // // live in paris, 1973 // mute bardo pond // conjunctio // volume nine // fire fire // four ways of dealing with one way // testament // rune grammofon ear infection // lying on a landline // monster energy cassette // hiccup aki takase and rudi mahall // hat and beard // duet for eric dolphy // yellowbird art pepper quintet // a bit of basie // smack up // contemporary yl featuring starker // good numbers // don't feed the pigeons // novelty ovrkast. // reset! // reset! ep // do more. erika de casier featuring they hate change // ice // still // 4ad tierra whack // shower song // world wide whack // interscope fats gaines band featuring zorina // born to dance // fats gaines band presents zorina // mad about bernice // dreaming of you // fader and friends volume one compilation // fader haha same // guess what to do // guess what to do 7" // sub pop the orchids // i just don't care // the way that you move 7" // pebble laetitia sadier // une autre attente // rooting for love // drag city soft covers // coming and going // soft serve // little lunch wicketkeeper // backwards again // zambroni // umpire

[yeoldehag]

The Zambroni Man

[ihaveimpeccabletaste]

he would be on a hockey team called the zambronis

[thadeeliv]

where you can find me if tumblr thanos snaps this blog

Twitter: @ZennecFox

deviantArt: zambrony

Pillowfort: zenpai

Discord: tiamonddiara#5167

Twitter is slightly more personal, deviantArt is strictly art (obvies), and Pillowfort is probly gonna just be the same as my tumblr is now.

[orangemoth]

can you imagine working at an ice rink with a mlp fan? they clean the ice w their zambrony 

[rosebowlhockey]

RBH defeated the Zambronies with a 5-1 win at RBH’s first Give Blood Play Hockey tourney 3 years ago. 2 Goals from Lance, and Dale, Koz and Jason each scored one to secure the win. That was a fun weekend! (at The Rinks - Irvine Inline) https://www.instagram.com/p/CGwFQobgHNJ/?igshid=1r3iht1u168qv

[willow-ink]

"Sup y'all little zambronies!"

- my brother, yelling from his room at nobody

[alemicheli76]

“Il sommo incantatore. Il richiamo della vendetta” di Francesco Zambroni, Dark Zone. A cura di Alessandra Micheli Cosa deve avere un fantasy per suscitare non tanto il mio interesse, quello parte solo alla parola libro, ma quel coinvolgimento particolare che presuppone la totale sospensione dell'incredulità e quel senso di estraniamento che capita con pochi libri?

[twighockeycompany]

Zambronies. Moves Like Jagr. There's some sweet team names here. Had to share.⠀ https://www.sportsfeelgoodstories.com/hockey-team-names/⠀ ⠀ #hockeylife #grabyourtwig #icehockey #hockey #teamnames #beerleague #ferda #hockeylove #odr #pondhockey #manager https://ift.tt/36rLF6N

[innerpenguinperson-blog]

Here it is, the video you’ve all been waiting for. My solution for anger issues. Vape up Zambronis.

[thousandmaths]

Pre-Introduction to Schur-Weyl Duality

This post is based very loosely on Mike Zambroni’s talk “From the Robinson-Schensted Algorithm to Schur-Weyl Duality” that he gave at our student combinatorics seminar. Unfortunately, I had a really hard time making sense of the Robinson-Schensted portion of that talk.

So instead I’m using it as an excuse to flesh out a particular detail that will show up in another talk post. It works pretty well for this, since it was the first time that I had really understood why the name “Schur-Weyl duality” makes any sense at all.

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Tensor Products

Schur-Weyl duality is, at its core, a statement about tensor products. Because tensor products are kind of annoying to describe properly, we’ll restrict our attention to the classical setting where we only need tensor products of vector spaces, which are really easy (well, at least, they’re at least no harder than vector spaces in general):

Definition. Let $W$ and $V$ be vector spaces with bases $w_1, w_2,\dots, w_m$ and $v_1, v_2, \dots, v_n$. Then the tensor product $V\otimes W$ is a vector space, whose basis elements we write as $w_i \otimes v_j$ for $1\leq i\leq m$ and $1\leq j\leq n$. In particular, $\dim(V\otimes W) = \dim(V)\cdot\dim(W)$.

It’s important to note that not every element in the tensor product looks like $v\otimes w$! In general, we are allowed linear combinations of these elements, such as $(0,1,0)\otimes (1,0,4) + (3,1,0) \otimes (1,1,4) \in \Bbb C^3\otimes \Bbb C^3$.

It will be convenient to introduce some extra notation: $V^{\otimes 2} = V\otimes V$ and similarly we define $V^{\otimes 3} = V\otimes V\otimes V$ and so on.

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Two Actions

We know there is a natural action of $\text{GL}(V)$, which is the set of all bijective linear transformations $\varphi:V\to V$ by simply evaluating the map $\varphi$ at each point. This gives rise to an equally natural action of $\text{GL}(V)$ on the tensor power $V^{\otimes k}$: since every one of its elements can be written 

$$\sum_{i_1,i_2,\dots, i_k=0}^\text{dim(V)} a_{i_1,i_2,\dots, i_k} v_{i_1} \otimes v_{i_2} \otimes \cdots \otimes v_{i_k}, $$

we simply apply $\varphi$ in as many places as it makes sense:

$$\sum_{i_1,i_2,\dots, i_k=0}^\text{dim(V)} a_{i_1,i_2,\dots, i_k} \varphi(v_{i_1}) \otimes \varphi(v_{i_2}) \otimes \cdots \otimes \varphi(v_{i_k}). $$

However, with the tensor product we suddenly have a new action on $V^{\otimes k}$ which doesn’t really make sense on just $V$. Namely, the symmetric group $S_k$ of permutations acts on this by simply shuffling around the factors in each term. So, using our example from before in $\Bbb C^3\otimes \Bbb C^3$, we find that the permutation 21 sends

$$ (0,1,0)\otimes (1,0,4) + (3,1,0) \otimes (1,1,4)  \qquad\mapsto\qquad  (1,0,4) \otimes (0,1,0) + (1,1,4) \otimes (3,1,0), $$

[ For those of you concerned with such things: the usual multiplication on $S_k$ turns this into a right-action, whereas clearly multiplication by a matrix is a left-action. We will be a little sloppy about this distinction (in no small part because we have never really been careful on this blog about exactly how the $S_k$ multiplication is defined). ]

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Schur-Weyl Duality

We may now give the classical statement of Schur-Weyl duality:

Theorem. Under the natural actions of $V^{\otimes k}$ by $\text{GL}(V)$ and $S_k$, it also has an action by $\text{GL}(V) \times S_k$. 

[ Alternatively, being careful about lefts and rights, $V^{\otimes k}$ is a $(\text{GL}(V),\Bbb Q[S_k])$-bimodule. ]

In other words, it doesn’t matter whether you permute order of the vectors and then apply $\varphi$, or whether you apply $\varphi$ and then permute the order of the vectors. 

Stated this way, you might wonder what the big deal is. The name makes sense now: the “duality” is referring to the fact that you can do things in either order (i.e. isomorphism after reversing the arrows). But you might be left scratching your head about why this thing even has a name. The answer lies in Schur’s remarkable observation that this “commutativity” extends all the way down to the level of irreducible decompositions.

Theorem (Schur-Weyl Duality). If $S^\lambda$ denotes the irreducible modules/representations of $S_k$, and $F^\lambda$ denotes the irreducible modules/representations of $\text{GL}(V)$, then 

$$ V^{\otimes k} \cong \bigoplus_{\lambda} m_{\lambda}(S^\lambda \otimes F^\lambda) $$

where the isomorphism is meant in the sense of $\text{GL}(V)\times S_k$ modules [or, bimodules].

The remarkable piece here is not so much that we can break things into irreducible components, but rather that in doing so we never have any “cross terms”. In other words, if $S^\lambda\otimes F^\mu$ shows up in the decomposition, then $\lambda=\mu$. At the big-picture level, we thus have this very deep connection between these two groups $S_n$ and $\text{GL}(V)$, which a priori don’t seem to have much to do with each other at all.

We conclude with a historical remark: Schur and Weyl have their names on this theorem for very different reasons. Schur was the one who proved the thing, in his thesis in 1905. Schur’s thesis, in fact, was sort of a beginning for abstract representation theory; in particular he was the one to define Frobenius reciprocity, which he named after his advisor. 

Frobenius, to be fair, was also quite interested in this circle of ideas, but it seemed they were relatively alone: even being passed over in Muir’s 1900 tome The Theory of Determinants in the Historical Order of Development. However, the physics community began to be interested in them for their applications in quantum mechanics, largely championed by Weyl. This spurred a revival of the topic among mathematicians; arguably Weyl’s influence is responsible for the central role of representation theory in the algebraic flavors of math today.

[cass-burger]

Zambrony.

[nihilismasscream-blog]

Liveblogin: Episode 23

This one’s one of my favorites! 

Tags: NihiDTMGLiveblog, Episode23

also, Zambronies.