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Lexi’s Raspberry Long Island Ice Tea was slid in front of her and Fay immediately took it.

Fay: Sexless for six months, I need this more than you do.

Lexi: That sounds like the beginning of a raunchy romance trilogy.

They both laughed and she glanced back at the handsome stranger. He didn’t look up this time and she turned back to face her slightly drunk friend. She was never one to approach guys. She was a self admitted awkward turtle. She’d rather play eye tag with him all night and leave without his number than to go over and say hi.

Lexi: I don’t think I can do it.

Fay: I’ll add two weeks on both sides of our bet if you go talk to himright now.

She grunted begrudgingly but got out of her seat. If he shot her down, she was definitely going to need the extra two weeks.

tagged by: @plutoandmarsmeetvenus

Name:Lexi

Nickname: Lex, savage candy bar

Gender: female

Star sign:virgo

Height: 5′4

Age:16

Hogwarts house: hufflepuff!

Favorite color(s): orange

Time right now: 3:53 PM

Hours of sleep: like eleven lmao i need to fix that

last thing I googled: is coconut oil good for your face?

Lucky number: 20

Favorite fictional character: Okay um definitely my Farkle. umm Hermione, Jim and Pam, Andy and April lol oh and my precious anime characters

Blanket I sleep with: its orange and it has the cutest patterns and i love it because its so comfortable

Favorite band/Artists: MELANIE MARTINEZ OMFG. um Ed Sheeran , secondhand serenade and my babies twenty one pilots

Dream Trip: Italy, or Utah because the mountains are beautiful

Dream job: writer, hands down.

What I’m wearing right now: a workout tank top and my pink pajamas with little sheep on them. idk why its about to be 4 pm

When did you make this blog: less than a month ago, one day when i couldn’t go any longer without reblogging all the amazing fanfics and hcs

Follower Count: 115

Posts: like 800 something

What do I post about: girl meets world stuff or other fandoms and ships, depending on what I’m obsessed with that day

Most active followers: idk i guess @shortstackhuckleberry and @giraffesarebetterthanyou and @callmecottoncandyface, @couldnt-think-of-a-funny-name @plutoandmarsmeetvenus

When did your blog reach its peak: never????

Why did you get tumblr: bc riarkle is endgame af

do you get asks on a regular basis: lately yeah because i’m in a freaking group chat lol

why did you choose your URL: because I AM RIARKLE TRASH (I’m surprised no one had taken it)

I CHOSE YOUUUU @callmecottoncandyface @giraffesarebetterthanyou @broadwaycorey @browncinderella13 @imoonrise25kingdom @riarkletopluto @shortstackhuckleberry @spookyflashlight @wizardofozshipper ,@tvshow4life and @omg-riarkle (((:

Introduction to Hecke Algebras

This talk was given by Michael Chmutov at the UMN 2016 summer Student Representation Theory Seminar. He cited Bourbaki’s *Lie Groups* Chap 4 Sec 2 [not free] and Mathas’ *Hecke Algebras and Schur Algebras of the Symmetric Group*.

The vast majority of this post will just be to motivate the definition of a Hecke algebra, which is, in no uncertain terms: weird. It will completely ignore the second half of the talk, which was about the cellularity of Hecke algebras. And, since there are lots of different loosely-related things called Hecke algebras, I should say that these are technically “Iwahori-Hecke” algebras.

[ I was… very surprised by the quality of this talk. Chmutov is a postdoc at UMN so I’ve seen him around, but I guess I had never heard a talk from him before. Previous SRTS talks haven’t been bad by any means, but this was a cut above the others I’ve seen. ]

——

We begin with the group $G=\text{GL}_n(\Bbb F_q)$, that is, the set of all invertible $n\times n$ matrices with coefficients in $\Bbb F_q$. [If this is unfamiliar, you can just think of this group as containing $n\times n$ tables of numbers between $0$ and $q$. This isn’t quite correct, but it’s good enough to follow the narrative.]

A Hecke algebra, at some very high level, is just a collection of functions $G\to\Bbb C$. You add functions in the Hecke algebra basically in the same way you add other functions, but multiplication is different, and morally it should be. That’s because ordinary function multiplication relies only on the ring structure of the codomain (here $\Bbb C$), but since there is also a multiplication structure in $G$, we want the Hecke algebra to recognize this somehow.

That recognition is achieved through **convolution**, a sort of glorified notion of “average”, which I’ll also write down formally: for two functions $X$ and $Y$, the convolution $X*Y$ is defined as

$$(X*Y)(g) = \frac{1}{|B|}\sum_{h\in G} X(h)Y(g^{-1}h),$$

where $B$ is the **Borel subgroup** of $G$, that is, the set of invertible upper-triangular matrices. [That is, you’re in the Borel subgroup if every number strictly below your top-left-to-bottom-right diagonal, is zero; and every number *on* that diagonal is *not* zero. The others can be whatever.]

If you’ve seen convolutions before, the $|B|$ probably looks a little weird. As far as I know, there’s no good reason you should expect to use $|B|$ instead of $|G|$; it simply because it makes the definition of the algebra prettier. (In particular, the relations will be independent of $n$, which would not be true if we used $|G|$.)

One of the advantages to using convolution is that it has very good properties with respect to $B$-invariance:

- If we only consider functions which are invariant on the cosets of $B$, that is, if $X(gb)=X(g)$ for every $g\in G$ and $b\in B$, then $X*Y$ is also invariant on the cosets of $B$.
- In fact, if we only consider the functions which are invariant on the
*double-*cosets of $B$, that is, if $X(b’gb)=X(g)$ for any $g\in G$ and $b,b’\in B$, then $X*Y$ is also invariant on the double-cosets of $B$.

With all this in mind, we can define the (Iwahori-)**Hecke algebra of type A**: it is the set of functions which are invariant on double-cosets of $B$, equipped with addition and convolution.

——

[ramp up]

But there’s more to the story. If you’ve wandered around in combinatorics circles (or this blog) for long enough, then you know that “type A” is code for “secretly there is a symmetric group here”. To find it, we’d have to venture into matrix factorizations. Because I don’t want to do that, I’ll just tell you the punchline, the so-called **Bruhat decomposition**:

$$G = \coprod_{w\in S_n} BwB.$$

When we write the product of a matrix and a permutation, we mean that the permutation is supposed to be represented as a permutation matrix. This means that each double-coset contains exactly one permutation matrix, and so each element in the Hecke algebra is completely determined by what it does to permutation matrices. Thus we have a “standard basis”:

$$ T_w(g) = \left\{ \begin{array}{ll} 1 & g\in BwB \\ 0 & \text{otherwise} \end{array} \right. $$

indexed by the symmetric group. Since the symmetric group is a Coxeter group, which has distinguished elements (the simple transpositions) $s_i$, this gives rise to distinguished elements of the Hecke algebra $T_i=T_{s_i}$. It’s natural to ask what happens if you convolve $T_w$ with $T_v$ for two elements of $S_n$. This turns out to be a bit complicated, so the next best thing is to ask what happens when you convolve $T_i$ and $T_j$.

Using “Coxeter group magic”, as Chmutov calls it, you first get that $T_i^2=qT_e+(q-1)T_i$. What you get for the others is even less pretty, but when you work it out, you end up with some commutativity relations which mirror those of $S_n$, namely $(T_iT_{i+1})^3=T_e$ and for non-adjacent indices, $(T_iT_j)^2=T_e$.

We now have removed all dependence on matrices from the story, which means we can at least define the **Hecke algebra of W**, for arbitrary Coxeter group $W$. It is a $\Bbb C$-algebra generated by $T_1,\dots, T_{n-1}$, and it has relations $T_i^2=qI+(q-1)T_k$ and $(T_iT_j)^{m_{ij}}=I$ for $i\neq j$, where $m_{ij}$ are the same as the powers in the presentation for $W$.

——

This construction can also be repeated over the $p$-adic numbers, $\Bbb Q_p$, with the price that $B$ must be replaced by the “Iwahori subgroup”, a distinctly number-theory-flavored object that I still find rather mysterious.

Top 5 biases

Tagged by @chanbaekaritz and thank you so much

5 - Jin, BTS

he reminds me of a bear and i fell in love with him because i saw him stuffing his face with food, like that was sexy af

4 - Lay/Yixing, EXO

dimples and thighs and adorableness and talent and lips and his laugh is just the cutest thing ever, this boy will never let me be loyal

3 - Sehun, Exo

I just love him more than chocolate

2 - Sehun, Exo

i don’t think he is real?

1 - Sehun, Exo

oh wow Sehun is definitely my number one, I don’t even need to think twice, sorry numbers 2 and 3

ps: I felt really bad for only choosing 5 biases so I decided to include another one that I really really love

Sehun from Exo

I couldn’t not mention him :)

I’m tagging @detectivetrabula @sebaekkk and @bisexualmeme

Tagged by @ace-murdock! (Thanks I love being tagged!)

Name: Alyssa

Nicknames: The Devil, One, Sis, Lys

Star sign: Taurus. I am a walking stereotype too

Favorite color: Blue. Always. I just love blue.

Gender: Female

Height: 5′10"

Average hours of sleep: Sleep Cycle says 8 which is a shame because I need 9-10 to function fully.

Last thing I googled: The definition for the word Skirmish

Number of blankets I sleep with: one generally but I can get up to three in the winter

Favorite fictional character: Ghirahim… I like his design a lot. Also pick a pokemon and chances are I love it. They count right?

Favorite celebrities: Brendon Urie, John Boyega, Daisy Ridley, Daniel Radcliffe, Emma Watson, Rami Male. All good.

Favorite book: lm sticking with Heir Apparent by Vivian Vande Velde, but I just read the Reckoned series by Brandon Sanderson and loved it so I am tempted to say that.

If you were to choose one fandom: This is tough… Um, Nintendo. (That way I can sneakily include Don’t Starve because it is on a Nintendo console)

The otp of otps: Lillicai I guess. I’m not huge into shipping but Lillicai is… I take it back. Psyren (Kreig x Maya) that ship is adorable.

Random fact: I am allergic to alcohol. Going to the doctor is constantly fraught with peril, and screw eating anything with vanilla in it.

Tagged: @davrial @lousylark @5007 @bjlemmart

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