Schur’s Lemma






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Representation of finite groups – notes in pictures

Of course, we have to find sub-structure within a structure – that’s what we do! I present before you, ladies and gentlemen, submodules.




And some new structure. How about building the vector space from the group? Group algebra – vector space + ring.


…and then ask the group to act on it.




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Representation of Finite Groups – Notes in pictures

A representation is simply a homomorphism between a group and the group of invertible matrices over a field, which is usually the field of complex numbers.



There are several approaches to developing the theory of representations, one of the popular ones is via FG-modules.



The connection between FG-modules and representations is simple – it is that of siamese twins.




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Basis of a vector space may not be strictly contained in it.

I thought that the basis must necessarily be smaller than the vector space itself. A colleague of mine gave me the following counterexample.
Let V be the direct sum of countably infinite copies of \mathbb{Z}_2.
V=\mathbb{Z}_2 \bigoplus \mathbb{Z}_2 \bigoplus \hdots=\{(x_1,x_2,\hdots)\mid x_i \in \mathbb{Z}_2, \text{only finitely many } x_i \neq 0. \}
Then cardinality of V is \aleph_0. Consider the set B defined as
B=\{(1,0,0,\hdots),(0,1,0,\hdots) \hdots\}.
B clearly forms a basis for V, but |B|=|V|=\aleph_0.

Of course such things cannot happen if finite dimensional vector spaces. One more proof that things get weird when they get to infinity.

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Notes from God’s own country

One of my teammates was tying the knot and we were cordially invited to grace the occasion. Since the venue happened to be in Kerela, plans were drawn, redrawn and then dropped. But Kerela did not give up on us; it beckoned us with renewed vigor and finally we had to give in. A few hasty notes scribbled on a hasty trip

Mumbai airport

I realize what’s keeping me air-borne is one man’s intelligence and his insistence on defying gravity. I trust this force as I board a plane or enjoy the backwaters on a motor boat. Men who made planes and boats and roads make being human something of a target for me. And it’s not because of what they did with their minds, but what they sought to do.

On the way from Kochi to Kottayam

Our driver is very chatty and instantly puts me at ease. Even though he’s speaking Malyalam and I obviously can’t follow a word, the sound of his incessant chatter is reassuring-it makes me feel welcomed. It’s past midnight but he’s not bothered about the lateness of the hour. We stop midway to drink black coffee from a roadside stall. Another day he brings us local liquor and serves us two courses in a glass kept in his car, I’m afraid for this very purpose. What a wonder words are, what powerful bridges! We ought to use them more lavishly. He poses for us as we click snaps and I’m wondering if all the people from this beautiful place are like him-simple, friendly, spontaneous. I know this is generalizing my limited experience. It is wrong, I realize, but it is human. I’m no patriot but I’ll remember this when I go to foreign land-not to bring bad name to the place I call home.

On the way to Munnar

I don’t want my snaps to be taken. This landscape minus me looks better. It looks better if I don’t look at the people.(Why is that, I am wondering)

There’s a waterfall on the way and we decide to play around a bit. Lot of people have stopped here. No matter how deep or shallow a person may be, how dumb or intelligent, this beauty spell bounds everyone. That, I think, is the mark of perfection.

For the past two days, we have done nothing but gaped in amazement over the hundreds of thousands of trees. What a place this is! The only embellishment it has to its merit is the green of the woods and the blue of the rivers and the shade of the azure skies. Yet, this alone is sufficient to qualify it as God’s own country for where else can you imagine God living? I wonder if it would be any less wonderful, any less magnificent if I saw more of it. I guess not. If it were not such a rare sight, I’d probably look at it less often but every time I did, I’m sure I’d be awestruck.


There’s a flower here which blooms once in twelve years and when it does, the entire valley is painted with its vibrant violet hue. I want to lay supine on these mountains and feel the roughness of their surface against my skin, like the coarse hands of a lover. What a combination of chance factors this is-that this beauty exists and that I can enjoy it-see the colors, feel the winds, smell the fragrance; that I have someone to miss I look at this; that I’ve loved and have been loved back. People who do not have one or all of these far outnumber people like me and I’m eternally grateful for every bit of it. I’m imbued with a desire to live, to see more of this and show it to the people I love, to write and sing and be happy. May all of this and more come true, for me and for everyone else. Amen!

For a while I am bothered by the fact that everything that I am thinking of and want to say has already been said by someone, somewhere. But I shrug off this feeling. What does it matter? How can it be helped if this sight takes everyone to the same heights or depths? What matters is that I’m here today and what it makes me feel must be said. Originality be damned-I must write this at the cost of sounding clichéd.

Cochin airport

I blow kisses to this lovely place and promise I’ll come back; next time around, not in such a haste. I will come here and let the fragrances of this place fill my senses, the quietness of this ring through me. Some other time, surely, when I have more time to stand and stare.

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Leicester Square

An atom consists of a heavy nucleus and light electrons revolving around it and spinning at the same time-much like the solar system, or the Milky Way, or any system in the cosmos. What a brilliant craftsman is God! From the atom to the cosmos, everything is same yet everything is different. The world is made up of many worlds, and Leicester square is made up of many, many worlds.

In the crowded square, a guy is kissing his girlfriend-not a passionate, desperate kiss but a deep yet tender kiss, soft as a song, as if the world revolved around these two. A girl who is drunk as a fish has her hands around the neck of the guy with her and is singing at the top of her voice-all guards down, she has probably never looked prettier than she does at the moment. A group of young Arab girls and boys is freaking out, drinking, smoking, and laughing. And a small town girl is looking at them and all the lights as if a frog has come out of the well. What could she be thinking? Maybe this-that things that were so difficult for her, if they were not so easy for others, life wouldn’t be so much fun. It is good that life is unfair and if that is good, then ‘why me’ is a very bad question.

Gosh, that sure is one long kiss. The guy has gently made his drunk girl sit near a telephone booth, maybe to fetch water or something. She is attracting a few disapproving frowns, but I say, what do these people know? What is the point in drinking if you are not getting drunk? Tomorrow morning when she wakes up with a headache, she will have sweet memories-of being drunk and of being taken care of when she was drunk. The guy has finally stopped kissing her and is whispering in her ears, surely the sweetest words, though of course I can’t hear them, nor do I want to. Being different is almost an obsession with me, but really, isn’t love the only idea anyone could possibly talk about-love of another person, love of life, absence of love, joy of love, pain of love?

A group of ISKCON members is chanting ‘Hare Rama, Hare Krishna’ on the streets of London, in the alleys of Leicester Square, where one practically breaths whiskey and beer. What drives these men and women? There is so much that I don’t understand-would books tell me all that I need to or want to know? I think yes. What is driving these men on this freezing, wet night? Maybe each of them, like me, is trying to make sense of this ‘crazy dream called life’.

An English lady in a saree with the most beatific smile on her face is asking the by-standers to join the group in the dance. Some of the people are contemptuous, some are puzzled, some are plain amused. But none of this matters to her. Her life is a never ending party and everyone is welcome to dance-the more, the merrier. The sight of a cycle rickshaw in London is a pleasure, almost comical-like the red light area of old Lucknow transported as is to Leicester square. The driver wants to make sure the passengers get their money’s worth and riding straight is the last thing on his mind.

A drunken teenager is pushed by his girl-friend in the middle of the group of dancing Krishna devotees. He is dancing as if he were in a discotheque and I am laughing my head off. He doesn’t even know he has made my day. Some people said that the universe is filled with ether-something no can see or smell or feel or weigh, but which was present in every inch of space where something else was not. I don’t think they were entirely wrong. There is something which ties us together, in ways beyond our comprehension or imagination. His pretty girl friend in a short skirt-short as short can be-joins him. It’s a scene straight from a Hollywood blockbuster. The frog from the well is watching enraptured. Funny thing is, neither this girl in the mini skirt nor this wide-eyed girl could ever imagine what the other is thinking. If I were the girl dancing on the streets, I’d be thinking of how I would narrate this incident to my friends tomorrow morning. And if not her, I would be thinking, I wish I could dance like her. Maybe I am bit of both, a bit if everyone on this square.

The group is passing through a Chinese dominated area-the Chinaland of London. Everywhere I turn my eyes I see people and more people, living miles away from home. Is it hope that has driven them out so far, or is it lack of hope? The old men smoking pipe and yacking away to glory seem to have all the time in the world. So often people say ‘memories came rushing’ but memories never come rushing-we always invite them, because everyone likes to live the memories.

A girl in the restaurant is sitting with her friend, quietly crying in her handkerchief. Why do I find even this beautiful? Why does this ugly, crowded square thrill me as much as the starry sky above? May be the alcohol has got to my head. Or maybe, I am not drunk-just high on life and on hope.

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It’s Tuesday morning after a long weekend and the pile of work staring me in the face tires me. I have no energy to trawl my way through it. The most innocuous mail irritates me, the smallest thing out of the plan I have for this day drives me nuts till I am swearing. Why this, I wonder? I am not so short-fused. The other day, when we were traveling to Scotland, just about nothing went according to the plan. We missed the bus, had to argue with the support staff no end to get alternate arrangements, wait in the biting cold for the bus which wouldn’t come and so on and so forth. But was I cool and smiling! Holidays are always welcome and get everyone charged up-that’s only fair. But life exists on weekends-I have a problem with that.

But then, such is life! On this Tuesday morning, when even typing is an effort, my mind goes back again and again to Edinburgh. I don’t believe in love at first sight but Edinburgh did sweep me off my feet. I always found London to be a city where the old and the new co-exist but Edinburgh was even better. Buildings made in probably the 18th century, cobbled pathway, ancient lamp-posts-that’s Edinburgh:Diagon Alley come alive. Almost instantly I like this city that wears her age with grace.
I am in the middle of the city and I don’t see a single new building. The old and the new don’t just co-exist in this city, they are intertwined so much so that they are inseparable-you can’t make out one from another and you just let their combination charm you.

My friends who had been to Scotland recently came back disappointed and told me that Edinburgh is just another big city. I want to travel the whole world but that comment made me inspect my whys. Is NY any different from London is any different from Shanghai is any different from Mallaig is any different from an Indian village? What exactly is it that I am expecting from a new place? Places like people, have souls. But I am on a three day grab-what-you-can tour through Scotland-do I expect her to reveal her soul to me? If I were Scotland, I would be offended by that expectation.

It was with these questions on my mind that I approached Edinburgh. But like a rich, ancient grandmother, she acts as a balm. Wealth has infinite shades-some subtly different, some glaringly so, different nevertheless. This place feels different from all other places I have been to…some people will tell me that this is only because I want to feel this way. That may be so, but who cares? It is what I feel, after all, that matters. So long as I feel it, this remains true.

We are traveling from Edinburgh to Mallaig from where we would take a ferry to the Isle of Skye. The train route from Fort William to Mallaig is said to be one of the ten most scenic routes in the world. As I look out from the window, I know that if anything can be more beautiful than this, surely I cannot imagine it. Comparison, I don’t think, would have been in the original scheme of things.
Its 8:45 PM and the sun doesn’t even hint at setting. This sure is going to be a long day. Every event since the of universe was meant to lead to this evening when I am traveling in the train and the sun stays up late so that I don’t miss a thing.

How do I describe this? I want to take this feeling back home-not its snaps, but what this scene makes me feel-to the people I love but how do I describe it? How do I tell them that there was this clear blue sky, and clouds scattered across it and that the shadow of the clouds painted the mountains and the shadow of the mountains painted the river and that the breeze created patterns on water-lightest ripples, infinitely similar yet infinitely different…how do I tell them that this makes me feel intensely alive and how do I explain why? I give up.

The mountains in the distance are covered with small white patches. We take bets on whether the patches are due to snow or water. It turns out to be snow. I remember someone telling me that during winters, the snow covered peaks are a sight to be seen. When these peaks are covered with snow, they are magic; when they are not, they are magic. When the water runs down their sides, we stop and stare at it; when it freezes in places, the sight is incredibly beautiful. This, what I am looking at now, is world in God’s own image and likeness. My deepest desire is to belong to this scene.

It’s past eleven in the night when we reach Mallaig. The sky is still lambent with a soft light, like a huge night lamp. Bed-time now!

Workplace seems to be riddled with landmines. I must watch every step lest I am blown up. You don’t even have to make a mistake to be dead-you just need to be; that’s reason enough to be killed. When I was standing by the sea in Scotland, I thought to myself that this is a place we read about in books or look at in postcards, and wish, without really wishing, we could go to. And here I am! I thought that when I am tired, frustrated I will think of this and there will be no yearning in me because I have been here, seen this. When I leave this place, I will not leave it behind. In my tired moments, I told myself, I will re-live this divine silence. I will listen again to the sound of bagpiper in Edinburgh and I will smile at the people lying face down on the grass without a care in the world. And that is what I am doing now-I am in Leavesden but I am in Mallaig, traveling to Isle of Skye, by a ferry.

It doesn’t matter where I focus my camera: every square inch is overflowing with beauty. Looking down at the deep blue ocean, I silently thank the people who have not thrown beer cans in the sea.

On the isle, I find more reasons to be thankful. Tranquility is a shade sea wears on some days, in some places for some people. It suits the ocean well and I am grateful for this.

The jagged rocks on the sea shore are strangely inviting. When I look at them closely, I notice that there are thousands of snails frozen dead on the rocks. How would this have happened, I wonder; and then this-these dead snails have become part of the rocks. The appearance that they render these rocks is unlike I have seen in any other place. The way it was meant to be, even death is beautiful. That which is not beautiful was not meant to be. Life without hope, sex without love, existence without meaning-these could not have been in the original scheme of things either.

Barely three miles down this rocky beach, we find a sandy beach. I have no doubt we can explain this, but can we?

I lie down on the sand. We are not really doing anything but we are having fun. This is perfection-when just being alive is fun; existence is a joy. Perfection is when Monday morning is like Friday evening and I believe that’s how it was meant to be. Life isn’t perfect yet but I will get there-slowly, steadily but surely.

Leavesden falls on the way to Scotland, though. Every place that I don’t want to be in falls on the way to the place I want to be in. And I want to be in every place that is on the way to the place I belong to. By God, living is fun.

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