Some interesting and elementary topics with connections to the representation theory?Fun applications of representations of finite groupsWhat role does the “dual Coxeter number” play in Lie theory (and should it be called the “Kac number”)?Gelfand-Tsetlin bases for Lie groups over finite fieldsLie group examplesHomotopical CombinatoricsAre convolution algebras ever “topologically noetherian”?Gauss, Jacobi, Kloosterman sums and representation theory in the $mathbb F_1$-worldReferences request: representations of classical groupsIs there an accessible exposition of Gelfand-Tsetlin theory?A second course in the representation theory
Some interesting and elementary topics with connections to the representation theory?
Fun applications of representations of finite groupsWhat role does the “dual Coxeter number” play in Lie theory (and should it be called the “Kac number”)?Gelfand-Tsetlin bases for Lie groups over finite fieldsLie group examplesHomotopical CombinatoricsAre convolution algebras ever “topologically noetherian”?Gauss, Jacobi, Kloosterman sums and representation theory in the $mathbb F_1$-worldReferences request: representations of classical groupsIs there an accessible exposition of Gelfand-Tsetlin theory?A second course in the representation theory
$begingroup$
I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (including the Jordan normal form, nilpotent operators, etc.), a real (and basics of complex) calculus, some elementary topology (basics from the general topology, fundamental groups and maybe something about manifolds or vector fields), basics of algebraic geometry and a lot of combinatorial stuff (including the graph theory and generating functions).
My main interest is the representation theory, so I'd like to discuss an algebraic topic connected with this branch of mathematics. It'll be cool if this topic contains some beautiful combinatorial constructions. Despite this, some level of abstraction is required... I think that it mustn't also be too famous (so the standard things like representations of symmetric groups (even in Vershik-Okounkov approach) or the basics of Lie theory aren't acceptable). Ideally, it has to be new to me...
Can someone give me a piece of advice about which topic can be chosen in this situation?
I thought about the cluster algebras (maybe in the flavor of the first pages of the paper [1]?)... But are there any elementary applications of them? (And, by the way, are there any classical monographs about this subject?)
[1] -- http://ovsienko.perso.math.cnrs.fr/Publis/FriezeNew1.pdf
UPD: There is a question which looks like similar to this on MO. Namely, Fun applications of representations of finite groups . But, of course, it's very different. The reason is that I'm more interested in representations of more complicated than finite groups structures like, for example, quivers or Lie algebras. So the other question isn't relevant: its topic is too narrow.
co.combinatorics rt.representation-theory soft-question mathematics-education cluster-algebras
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
$endgroup$
|
show 1 more comment
$begingroup$
I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (including the Jordan normal form, nilpotent operators, etc.), a real (and basics of complex) calculus, some elementary topology (basics from the general topology, fundamental groups and maybe something about manifolds or vector fields), basics of algebraic geometry and a lot of combinatorial stuff (including the graph theory and generating functions).
My main interest is the representation theory, so I'd like to discuss an algebraic topic connected with this branch of mathematics. It'll be cool if this topic contains some beautiful combinatorial constructions. Despite this, some level of abstraction is required... I think that it mustn't also be too famous (so the standard things like representations of symmetric groups (even in Vershik-Okounkov approach) or the basics of Lie theory aren't acceptable). Ideally, it has to be new to me...
Can someone give me a piece of advice about which topic can be chosen in this situation?
I thought about the cluster algebras (maybe in the flavor of the first pages of the paper [1]?)... But are there any elementary applications of them? (And, by the way, are there any classical monographs about this subject?)
[1] -- http://ovsienko.perso.math.cnrs.fr/Publis/FriezeNew1.pdf
UPD: There is a question which looks like similar to this on MO. Namely, Fun applications of representations of finite groups . But, of course, it's very different. The reason is that I'm more interested in representations of more complicated than finite groups structures like, for example, quivers or Lie algebras. So the other question isn't relevant: its topic is too narrow.
co.combinatorics rt.representation-theory soft-question mathematics-education cluster-algebras
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
$endgroup$
1
$begingroup$
may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials.
$endgroup$
– vidyarthi
11 hours ago
1
$begingroup$
also may be something about using representation theory in graph coloring
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...
$endgroup$
– kotlinski
11 hours ago
$begingroup$
Possible duplicate of Fun applications of representations of finite groups
$endgroup$
– Francois Ziegler
10 hours ago
$begingroup$
@FrancoisZiegler I've edited my question especially for you.
$endgroup$
– kotlinski
9 hours ago
|
show 1 more comment
$begingroup$
I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (including the Jordan normal form, nilpotent operators, etc.), a real (and basics of complex) calculus, some elementary topology (basics from the general topology, fundamental groups and maybe something about manifolds or vector fields), basics of algebraic geometry and a lot of combinatorial stuff (including the graph theory and generating functions).
My main interest is the representation theory, so I'd like to discuss an algebraic topic connected with this branch of mathematics. It'll be cool if this topic contains some beautiful combinatorial constructions. Despite this, some level of abstraction is required... I think that it mustn't also be too famous (so the standard things like representations of symmetric groups (even in Vershik-Okounkov approach) or the basics of Lie theory aren't acceptable). Ideally, it has to be new to me...
Can someone give me a piece of advice about which topic can be chosen in this situation?
I thought about the cluster algebras (maybe in the flavor of the first pages of the paper [1]?)... But are there any elementary applications of them? (And, by the way, are there any classical monographs about this subject?)
[1] -- http://ovsienko.perso.math.cnrs.fr/Publis/FriezeNew1.pdf
UPD: There is a question which looks like similar to this on MO. Namely, Fun applications of representations of finite groups . But, of course, it's very different. The reason is that I'm more interested in representations of more complicated than finite groups structures like, for example, quivers or Lie algebras. So the other question isn't relevant: its topic is too narrow.
co.combinatorics rt.representation-theory soft-question mathematics-education cluster-algebras
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
$endgroup$
I'm going to give a talk to talented high school seniors (for nearly 1.25-1.75 hours, maybe a little bit longer). They know some abstract algebra (groups, rings, modules...), linear algebra (including the Jordan normal form, nilpotent operators, etc.), a real (and basics of complex) calculus, some elementary topology (basics from the general topology, fundamental groups and maybe something about manifolds or vector fields), basics of algebraic geometry and a lot of combinatorial stuff (including the graph theory and generating functions).
My main interest is the representation theory, so I'd like to discuss an algebraic topic connected with this branch of mathematics. It'll be cool if this topic contains some beautiful combinatorial constructions. Despite this, some level of abstraction is required... I think that it mustn't also be too famous (so the standard things like representations of symmetric groups (even in Vershik-Okounkov approach) or the basics of Lie theory aren't acceptable). Ideally, it has to be new to me...
Can someone give me a piece of advice about which topic can be chosen in this situation?
I thought about the cluster algebras (maybe in the flavor of the first pages of the paper [1]?)... But are there any elementary applications of them? (And, by the way, are there any classical monographs about this subject?)
[1] -- http://ovsienko.perso.math.cnrs.fr/Publis/FriezeNew1.pdf
UPD: There is a question which looks like similar to this on MO. Namely, Fun applications of representations of finite groups . But, of course, it's very different. The reason is that I'm more interested in representations of more complicated than finite groups structures like, for example, quivers or Lie algebras. So the other question isn't relevant: its topic is too narrow.
co.combinatorics rt.representation-theory soft-question mathematics-education cluster-algebras
co.combinatorics rt.representation-theory soft-question mathematics-education cluster-algebras
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
edited 6 hours ago
kotlinski
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
asked 11 hours ago
kotlinskikotlinski
1635 bronze badges
1635 bronze badges
1
$begingroup$
may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials.
$endgroup$
– vidyarthi
11 hours ago
1
$begingroup$
also may be something about using representation theory in graph coloring
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...
$endgroup$
– kotlinski
11 hours ago
$begingroup$
Possible duplicate of Fun applications of representations of finite groups
$endgroup$
– Francois Ziegler
10 hours ago
$begingroup$
@FrancoisZiegler I've edited my question especially for you.
$endgroup$
– kotlinski
9 hours ago
|
show 1 more comment
1
$begingroup$
may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials.
$endgroup$
– vidyarthi
11 hours ago
1
$begingroup$
also may be something about using representation theory in graph coloring
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...
$endgroup$
– kotlinski
11 hours ago
$begingroup$
Possible duplicate of Fun applications of representations of finite groups
$endgroup$
– Francois Ziegler
10 hours ago
$begingroup$
@FrancoisZiegler I've edited my question especially for you.
$endgroup$
– kotlinski
9 hours ago
1
1
$begingroup$
may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials.
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials.
$endgroup$
– vidyarthi
11 hours ago
1
1
$begingroup$
also may be something about using representation theory in graph coloring
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
also may be something about using representation theory in graph coloring
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...
$endgroup$
– kotlinski
11 hours ago
$begingroup$
@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...
$endgroup$
– kotlinski
11 hours ago
$begingroup$
Possible duplicate of Fun applications of representations of finite groups
$endgroup$
– Francois Ziegler
10 hours ago
$begingroup$
Possible duplicate of Fun applications of representations of finite groups
$endgroup$
– Francois Ziegler
10 hours ago
$begingroup$
@FrancoisZiegler I've edited my question especially for you.
$endgroup$
– kotlinski
9 hours ago
$begingroup$
@FrancoisZiegler I've edited my question especially for you.
$endgroup$
– kotlinski
9 hours ago
|
show 1 more comment
2 Answers
2
active
oldest
votes
$begingroup$
Here three suggestions:
-An applications of representation theory to chemistry:
"The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub
-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.
-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .
For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
$endgroup$
add a comment
|
$begingroup$
I gave a 4 minute(!) talk for an audience of high-school students about my research.
I talked about the cyclic sieving phenomenon, which has a close connection with representation theory.
Most people have an intuitive idea about what symmetries are, and what objects are "more symmetric", so this concept is not that difficult.
Here is a link to the slides (swedish).
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
$endgroup$
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
add a comment
|
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Here three suggestions:
-An applications of representation theory to chemistry:
"The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub
-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.
-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .
For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
$endgroup$
add a comment
|
$begingroup$
Here three suggestions:
-An applications of representation theory to chemistry:
"The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub
-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.
-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .
For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
$endgroup$
add a comment
|
$begingroup$
Here three suggestions:
-An applications of representation theory to chemistry:
"The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub
-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.
-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .
For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
$endgroup$
Here three suggestions:
-An applications of representation theory to chemistry:
"The Representation-Theory for Buckminsterfullerene" by Gordon James: https://www.sciencedirect.com/science/article/pii/S0021869384712130?via%3Dihub
-For a mix with probability theory the book Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains by Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli is great and gives insight to the works of Persi Diaconis.
-The applications of the representation theory of the symmetric group in Hurwitz theory, see for example the book Riemann Surfaces and Algebraic Curves: A First Course in Hurwitz Theory by Renzo Cavalieri and Eric Miles .
For cluster algebras there is a forthcoming book by Sergey Fomin, Lauren Williams and Andrei Zelevinsky : https://arxiv.org/abs/1608.05735 .
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
edited 10 hours ago
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
answered 10 hours ago
MareMare
4,8333 gold badges14 silver badges39 bronze badges
4,8333 gold badges14 silver badges39 bronze badges
add a comment
|
add a comment
|
$begingroup$
I gave a 4 minute(!) talk for an audience of high-school students about my research.
I talked about the cyclic sieving phenomenon, which has a close connection with representation theory.
Most people have an intuitive idea about what symmetries are, and what objects are "more symmetric", so this concept is not that difficult.
Here is a link to the slides (swedish).
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
$endgroup$
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
add a comment
|
$begingroup$
I gave a 4 minute(!) talk for an audience of high-school students about my research.
I talked about the cyclic sieving phenomenon, which has a close connection with representation theory.
Most people have an intuitive idea about what symmetries are, and what objects are "more symmetric", so this concept is not that difficult.
Here is a link to the slides (swedish).
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
$endgroup$
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
add a comment
|
$begingroup$
I gave a 4 minute(!) talk for an audience of high-school students about my research.
I talked about the cyclic sieving phenomenon, which has a close connection with representation theory.
Most people have an intuitive idea about what symmetries are, and what objects are "more symmetric", so this concept is not that difficult.
Here is a link to the slides (swedish).
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
$endgroup$
I gave a 4 minute(!) talk for an audience of high-school students about my research.
I talked about the cyclic sieving phenomenon, which has a close connection with representation theory.
Most people have an intuitive idea about what symmetries are, and what objects are "more symmetric", so this concept is not that difficult.
Here is a link to the slides (swedish).
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
answered 5 hours ago
Per AlexanderssonPer Alexandersson
7,9297 gold badges46 silver badges89 bronze badges
7,9297 gold badges46 silver badges89 bronze badges
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
add a comment
|
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
$begingroup$
A great material! Thank you!
$endgroup$
– kotlinski
5 hours ago
add a comment
|
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1
$begingroup$
may be something along the lines of polya enumeration, like polya polynomial and its relation to symmetric polynomials, including schur polynomials.
$endgroup$
– vidyarthi
11 hours ago
1
$begingroup$
also may be something about using representation theory in graph coloring
$endgroup$
– vidyarthi
11 hours ago
$begingroup$
@vidyarthi Thank you for your comment! But I think that the (i) more abstract and more algebraic, (ii) more representation-theoretic topics are required. But the idea about the representation theory in graph coloring looks interesting. I'll think about something like this...
$endgroup$
– kotlinski
11 hours ago
$begingroup$
Possible duplicate of Fun applications of representations of finite groups
$endgroup$
– Francois Ziegler
10 hours ago
$begingroup$
@FrancoisZiegler I've edited my question especially for you.
$endgroup$
– kotlinski
9 hours ago