In the coffee hour discussion today, Nick gave me an interesting explicit example of an exceptional isomorphism of Lie groups.

G = \mathbb{SL}(4, \mathbb C) \to \mathbb{SO}(6, \mathbb C)

is an isomorphism via the \bigwedge^2 map. Let me elaborate. The group G acts naturally on V = \mathbb C^4. If \{e_1, e_2, e_3, e_4\} is a basis of V, then a basis of \bigwedge^2 V could be taken as \{ e_1 \wedge e_2, e_1 \wedge e_3, \cdots, e_3 \wedge e_4 \}. Thus, \mathbb{GL}(\bigwedge^2 V) is \mathbb {GL}(6, \mathbb C) and our G-action gives a monomorphism of groups:

\bigwedge^2 : G=\mathbb{SL}(4,\mathbb C) \to \mathbb{GL}(\bigwedge^2 V) = \mathbb {GL}(6,\mathbb C).

The image must actually be inside \mathbb{SL}(6,\mathbb C).

The quadratic form

(\; , \; ) : \bigwedge^2 V \times \bigwedge^2 V \to \mathbb C,

(e_i \wedge e_j , e_k \wedge e_l ) \mapsto e_i \wedge e_j \wedge e_k \wedge e_l

is symmetric (two permutations) and preserves the G– action, i.e.,

(g . \left(e_i \wedge e_j\right) , \left( g . e_k \wedge e_l \right) ) = (e_i \wedge e_j, e_k \wedge e_l) .

Hence the image of G in \mathbb{GL}(6, \mathbb C) \simeq \mathbb{GL}(\bigwedge^2 V) must also preserve this symmetric bilinear form. Thus, \bigwedge^2(G) \subseteq \mathbb{SO}(\bigwedge^2 V). By dimension consideration, they must be equal.

I ran the risk of opening a pandora’s box if I posted this on Facebook (with random ‘friends’ starting discussions on my timeline) so I decided to post my views here. This post is for me to walk down the memory lane by reading it in the future rather than start a fight here. Plus, it’s been a notoriously long time since I posted anything here.

Modi was unanimously elected the Prime Minister in May 2014 when BJP won the elections with an overwhelming majority. With over an year of the government being in power, issues are being raised about the growing intolerance. There are two things I would like to point out.

(1) Everything Modi and the BJP does is being scrutinized. You either love Modi or hate him; there isn’t a middle ground. Why didn’t people focus on the Congress and it’s wrongdoings? Incidents like the Dadri lynching are outrageous and bring shame to my country but they haven’t cropped up only after the BJP rose to power, they only became more visible. The problem with Indian secularism is that it works only in favour of minorities. The fact that media is no longer unbiased in it’s reporting and goes only for sensationalism doesn’t help either.

(2) As citizens with the power of (just) “one” vote, if good governance is our objective, we should vote for the party with ‘lesser evil’. That automatically gives the onus on politicians to perform. If BJP doesn’t live up to it’s standards, vote it out! If it does improve my country, re-elect Modi. With a highly active and involved ministry, I think (hope) Modi’s cabinet gets another chance. After all, don’t we all want the Uniform Civil Code re-implemented, as was promised in the BJP manifesto?

I was wondering the rationale behind naming parabolic subgroups of linear algebraic groups. The answer, interestingly, comes from the action of SL(2,\mathbb R) on the upper half plane. (I came up with this little discovery on my own.)

The orbit of the point i under the action of the standard parabolic subgroup of SL(2,\mathbb R) is a parabola.

The upper half-plane is an object that comes up in many parts of mathematics, hyperbolic geometry, complex analysis and number theory to name a few. The group SL(2,\mathbb R) acts on it by fractional linear transformations:

\begin{pmatrix} a & b \\ c & d \end{pmatrix} .\ z \mapsto \displaystyle \left( \frac{az+b}{cz+d} \right) .

A parabolic subgroup is it’s subgroup P such that quotienting by P gives a compact variety. Upto conjugation, the only parabolic subgroup of SL(2,\mathbb R) is \begin{pmatrix} * & * \\ 0 & * \end{pmatrix}. It’s action on the point i is given by:

\begin{pmatrix} x & y \\ 0 & x^{-1} \end{pmatrix} . i = \displaystyle \left( \frac{xi+y}{0i+x^{-1}} \right) = y + i x^2

whose locus is a parabola.

I wonder why textbooks in algebraic groups don’t mention this!

EDIT: (21 April, 2015) The above calculation is WRONG. I don’t know the answer to “parabolic”.

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Now I’ll think twice before going to Greyhouse. Here is why.

I went to the coffee-house today evening for a cup of GreyCap and Cogdell’s book on Automorphic L-functions. Greyhouse is a wonderful place to do some Mathematics and at times, just gaze around and see what other people are doing. My Chess buddies often meet here for some rapid games. Today the place was less crowded since the school is closed for summer.

A lady approaches me and asks me my age. I have seen her earlier around the campus and even in Greyhouse. I remember last summer, she once interrupted our Chess discussion and made an awkward conversation. Kerem pointed out she’s crazy but “not guilty until proven” is my principle.

Not wanting to create a fuss, I told her my approximate age and she left me alone. After a while, she came to me and asked if I could share the table with her. Honouring the etiquette of the coffee-house, I welcomed her. Now she asks if I could be her friend. I say yes. But soon things start getting weird.

“Would you like to be my friend with benefits?”

“No.” (What the fcuk!!)

“Do you know what that means?”

“I think I know.”

“Hugging, kissing, … ”

“Yes I know” (This was getting creepy)

“Do you have a girlfriend?”

“No”

“Why?”

“Because I love some things (pointing at my book) more than others.”

She laughed aloud enough for everyone around to hear. I could feel the pain hidden in that laughter. I felt pity toward her.

She tore off a page from her pocket-diary and wrote her phone number on it. When I accepted it, she asked mine. I refused to share my cell number with a creepy lady I just met. She asked her note back and I gladly obliged. With some more awkward comments, she left. I shared a moment with the Indian girl sitting at the adjacent table; I could tell she overheard our conversation.

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The conversation had an impact on me I hadn’t imagined. It was not just any other crazy experience but somewhat traumatizing one. I felt shaken, mildly violated although all she did was have an extremely weird conversation. I wonder what it would be to be a girl in India. Eve-teasing, stalking, groping are real problems because of roadside Romeos – jokers who don’t know how to woo a girl but know a lot of other creepy mischief. The askew sex-ratio also doesn’t help. My (female) friend used to tell me the little ways she used to retaliate against these miscreants. They should be brought to book. Hopefully “Ab ki bar Modi sarkar” or AAP improve things.

पूजा: अाईला सांगू नकोस!

I came across a simple statement in finite group theory that I’m almost upset no one told me earlier. The source is Serre’s book ‘Linear Representations of Finite Groups’. Serre used this statement below to define / prove Brauer’s theorem on induced representations of finite groups, using which one proves the meromorphicity of Artin-L-functions. Here it goes.

G is a finite group and p is a fixed prime. An element x of G is called punipotent if x has order a power of p and pregular if it’s order is prime to p.

Cool result:  Every element x in G can be uniquely written as

x = x_u x_r;

where

  • x_u is p-unipotent, x_r is p-regular,
  • x_u and x_r commute and
  • they are both powers of x.

The proof is really easy. Just replace G by the (finite) cyclic group generated by x!

I call it the Jordan decomposition because we have a similar decomposition for endomorphisms (among other things).

Let V be a finite dimensional vector space over an algebraically closed field of characteristic zero (just in case!). Each x \in \text{End }(V) can be uniquely written as

x = x_s x_u;

where

  • x_s is semisimple (diagonalizable), x_u is nilpotent,
  • they both commute and
  • they are polynomials in x without a constant term.

Pretty cool, huh!

This post is just a collection of basic results I have compiled for referring to in desperate times Nothing too deep except Haar-von Neumann’s theorem.

A measure \mu on a locally compact (Hausdorff, always) group G is left-invariant if

\boxed{ \int_G f(s^{-1}x) \text{d} \mu(x) = \int_G f(x) \text{d} \mu(x). }

The most important theorem in this topic is the existence and uniqueness of the Haar measure (proven by Haar and von Neumann, respectively).

Theorem 1: On every locally compact group G there is a unique (up to a positive constant of proportionality) left-invariant positive measure \mu \neq 0.

Proposition 2: G, \mu as usual. For G to have a finite measure, it is necessary and sufficient that G be compact!

Proposition 3: There is a continuous group-homomorphism (called right modulus), \triangle_r : G \to \mathbb R^*_+ such that

f(xs^{-1}) \text{d} \mu(x) = \triangle_r(s) \int f(x) \text{d} \mu(x).

Proposition 4: Let \mu (respectively, \nu) be a left-invariant Haar measure on locally compact groups H (resp., K). Then the product integral on G = H \times K is left-invariant and

\boxed{\triangle^G_r(s,t) = \triangle^H_r(s) . \triangle^K_r(t).}

Corollary 5: G is unimodular (i.e., \triangle_r =1) precisely when H and K are so.

The most important example (for me) is the general linear group over number fields or p-adics and its subgroups. All semisimple (and more generally reductive) groups are unimodular. Compact groups are unimodular. Abelian groups are trivially so. However the measure on Borel (and parabolic) subgroups (i.e., upper triangular matrices) is not unimodular. The above proposition 4 allows one to transfer the Levi decomposition on the groups to their measures.

I just finished watching The Silence of The Lambs (1991).  Starring Jodie Foster and Anthony Hopkins, it is probably the first of its kind and has won many Academy Awards. The mystery thriller is rated 8.6/10 on IMDB so you can imagine how amazing it would be.

It might be just me but I find an uncanny resemblance with The Call (2013) starring Halle Berry. I watched it yesterday; the story is about a 911 call. Both movies involve psychopath killers, a kidnapped girl and a female cop. In The Silence …, Jodie Foster is an FBI agent whereas Halle Berry is a 911 operator. Both movies are centered around rescuing the victim. Both movies end with cops being misled to the wrong location whereas the actress locates the right house, barges in alone without any reinforcements, is about to be killed by the perpetrator but finally manages to overpower the bad guy!

Too much for coincidence? !!

————————————————————————

Tailpiece: I just came across the most widely spoken constructed international auxiliary language – Esperanto. http://en.wikipedia.org/wiki/Esperanto

It has been almost a week since I returned to Purdue. The past month in India passed so quickly it just felt like a week. I was glad I could meet my family and so many friends.

I attended a workshop and conference on Galois representations at the TIFR. Situated besides the sea-shore, it always feels great to visit the Tata Institute. I was very surprised Prof. Ghate recognized me although we just met  once, that too three years ago. I met Prof. Rajan (my VSRP mentor) and my Masters advisor Prof. Anandavardhanan. It was nice to finally meet Sandeep Verma, a student of my advisor Dr. Shahidi. I couldn’t meet Anand Sawant since he was making an academic visit to Germany but it was good to talk to Sachin Sharma, my old friend at IMSc doing his post-doc at Tata. Although I couldn’t meet Arghya, I made friends with his friends – Shaunak, Aditya, Vineet and Ashay – and we discussed some Mathematics. There were prominent mathematicians attending the conference including Marie-France Vigneras, Sujatha Ramadorai, Pierre Colmez, Dipendra Prasad and Chandrashekhar Khare.

http://conferences.math.tifr.res.in/photos/llc2013/

Conference photo

Back home, it was after a long three-year wait that my parents, sister and I were together. We celebrated by going on a short vacation to Jaisalmer and Jodhpur in Rajasthan. There were sand dunes around (it’s a freaking desert) and we had a camel safari. It was fun!

The vibrant colours of India

The vibrant colours of India (cloth shop inside Jodhpur fort)

Rajasthani folk dancer (Yes, they are glasses she's balancing on)

Rajasthani folk dancer (Yes, they are glasses she’s balancing on)

On the pretext of giving his iPad to his folks, I met Partha’s dad.  Fortunately he didn’t ask me usual questions Indian parents ask! Shashank is one of my best friends and he too came to India to attend Avdhut’s marriage. We went to visit Vaibhav (photo below) and had an interesting discussion on genetics and evolution. As always, it was fun meeting Amar – my roomie at IITB. He went paragliding near Pune a day before I left for the US and seeing these pics, I regret not being able to join. Yes, I also met my Purdue roommate Nikhil, who came all the way from Pune to meet me. Nikhil, Rubin and I had a great time cracking Purdue jokes here in India. Another close friend Ameya married last year and in this India visit, I met Asmita and him. (Everyone seems to be getting married😦 I’m getting older.)

Vaibhav and Shashank

Vaibhav and Shashank

The star of the show was Avdhut, whose marriage was a pushing reason I visited India. His brothers had arranged a bachelors’ party a week before the wedding and I won’t go into details because what happens in Murud stays in Murud. But yes, I did meet Arun and have curious philosophical discussions with him. Avdhut’s marriage was a good excuse for VNITians to meet up – Raju, Tapa, Swapnil, Milind, SKS, Khan and the great Maythegod! Bless the newly wed couple!

Avdhut weds Aasawari

Avdhut weds Aasawari

My awesome dream-visit was brought down to earth after landing at Chicago. It was -27 degree Celcius cold!

Tailpiece:

* Just yesterday, I got a Macbook Pro! I promised myself I’d write this post on my new Mac.

* I was stuck up in Chicago since the interstate I-65 was dysfunctional due to heavy snowfall; shuttles were canceled and the train was overbooked. Kyle rescued me by giving a ride since he too was driving from Chicago back to WL. Hail Facebook!

Kyle gave me a ride to Purdue

Kyle gave me a ride to Purdue

* Steve Spallone is visiting Purdue next week. He is an American mathematician working in India and I’m quite the opposite. We discovered each others’ blogs accidently. Curiously, he too works in number theory and in fact is Shahidi’s collaborator. I’m looking forward to his talk!

The land of organized chaos

The land of organized chaos

Let G = GL_2(F) for some field F and B be the subgroup of upper triangular matrices in G. As usual, let \mathbb P^1 = \mathbb P^1(F) denote the projective plane over F.

As F-varieties, G/B is isomorphic to \mathbb P^1 because B is the stabilizer of the usual (Mobius) action of G on \mathbb P^1. The resulting action of G/B on \mathbb P^1 is 1-transitive and fixed-point free. Indeed, counting points when F is a finite field with q elements agrees with the facts that

\# G(\mathbb F_q) = (q^2-1)(q^2-q)

and

\# \mathbb P^1(\mathbb F_q) = q+1.

This generalizes in that parabolic subgroups (closed subgroup containing a Borel subgroup, i.e., containing the upper triangular matrices upto conjugation, for the general linear group) of an algebraic group give rise to projective varieties. They are precisely the stabilizers of a flag and conversely, the set of flags can be given the structure of an F-space and in particular a topology wherein they are compact spaces (being projective spaces). They are called flag varieties.

About me

Abhishek Parab

I? An Indian. A mathematics student. A former engineer. A rubik's cube addict. A nature photographer. A Pink Floyd fan. An ardent lover of Chess & Counter-Strike.

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