representations of cherednik algebras - mit mathematics€¦ · representations of cherednik...
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![Page 1: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/1.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Representations of Cherednik Algebras
Matthew LipmanMentor: Gus Lonergan
Sixth Annual MIT-PRIMES ConferenceMay 21, 2016
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 2: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/2.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
One nice set of operators are those d which can be written interms of differentiation and multiplication by fixed polynomials,known as differential operators.
The operator A = x ∂∂x (take the derivative, then multiply by x) is
one. However, the operator S12 that replaces all x1’s with x2’s andvice-versa, is not.
The product rule yields ∂∂x x = x ∂
∂x + 1 and ∂∂x2
x1 = x1∂
∂x2.
Consider the algebra generated by x1, x2, . . . , xn (multiplication byvariables) and ∂1, ∂2, . . . , ∂n (differentiation), subject to[xi , ∂j ] = xi∂j − ∂jxi = −δij , i.e. −1 if i = j and 0 otherwise.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 3: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/3.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
One nice set of operators are those d which can be written interms of differentiation and multiplication by fixed polynomials,known as differential operators.
The operator A = x ∂∂x (take the derivative, then multiply by x) is
one. However, the operator S12 that replaces all x1’s with x2’s andvice-versa, is not.
The product rule yields ∂∂x x = x ∂
∂x + 1 and ∂∂x2
x1 = x1∂
∂x2.
Consider the algebra generated by x1, x2, . . . , xn (multiplication byvariables) and ∂1, ∂2, . . . , ∂n (differentiation), subject to[xi , ∂j ] = xi∂j − ∂jxi = −δij , i.e. −1 if i = j and 0 otherwise.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 4: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/4.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
One nice set of operators are those d which can be written interms of differentiation and multiplication by fixed polynomials,known as differential operators.
The operator A = x ∂∂x (take the derivative, then multiply by x) is
one. However, the operator S12 that replaces all x1’s with x2’s andvice-versa, is not.
The product rule yields ∂∂x x = x ∂
∂x + 1 and ∂∂x2
x1 = x1∂
∂x2.
Consider the algebra generated by x1, x2, . . . , xn (multiplication byvariables) and ∂1, ∂2, . . . , ∂n (differentiation), subject to[xi , ∂j ] = xi∂j − ∂jxi = −δij , i.e. −1 if i = j and 0 otherwise.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 5: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/5.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Recall A = x ∂∂x = ∂
∂x x − 1.
We have
A2 =x ∂
∂x x ∂
∂x
=x(x ∂
∂x + 1) ∂∂x
=x2 ∂2
∂2x + x ∂
∂x
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 6: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/6.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Recall A = x ∂∂x = ∂
∂x x − 1.
We have
A2 =x ∂
∂x x ∂
∂x
=x(x ∂
∂x + 1) ∂∂x
=x2 ∂2
∂2x + x ∂
∂x
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 7: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/7.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Recall A = x ∂∂x = ∂
∂x x − 1.
We have
A2 =x ∂
∂x x ∂
∂x
=x(x ∂
∂x + 1) ∂∂x
=x2 ∂2
∂2x + x ∂
∂x
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 8: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/8.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Recall A = x ∂∂x = ∂
∂x x − 1.
We have
A2 =x ∂
∂x x ∂
∂x
=x(x ∂
∂x + 1) ∂∂x
=x2 ∂2
∂2x + x ∂
∂x
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 9: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/9.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Now, suppose we want to extend this to have, e.g. S12. We caneasily apply a similar logic to add all permutations into our algebra.
The Dunkl operators Di =∂
∂xi− c
∑j 6=i(xi − xj)
−1(1− Sij)commute with each other (with c a fixed constant).They also sharethe familiar property of derivatives that deg Dp ≤ (deg p)− 1.
It is apparently useful in certain fields, like the representationtheory of Sn.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 10: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/10.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Now, suppose we want to extend this to have, e.g. S12. We caneasily apply a similar logic to add all permutations into our algebra.
The Dunkl operators Di =∂
∂xi− c
∑j 6=i(xi − xj)
−1(1− Sij)commute with each other (with c a fixed constant).
They also sharethe familiar property of derivatives that deg Dp ≤ (deg p)− 1.
It is apparently useful in certain fields, like the representationtheory of Sn.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 11: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/11.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Now, suppose we want to extend this to have, e.g. S12. We caneasily apply a similar logic to add all permutations into our algebra.
The Dunkl operators Di =∂
∂xi− c
∑j 6=i(xi − xj)
−1(1− Sij)commute with each other (with c a fixed constant).They also sharethe familiar property of derivatives that deg Dp ≤ (deg p)− 1.
It is apparently useful in certain fields, like the representationtheory of Sn.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 12: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/12.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Now, suppose we want to extend this to have, e.g. S12. We caneasily apply a similar logic to add all permutations into our algebra.
The Dunkl operators Di =∂
∂xi− c
∑j 6=i(xi − xj)
−1(1− Sij)commute with each other (with c a fixed constant).They also sharethe familiar property of derivatives that deg Dp ≤ (deg p)− 1.
It is apparently useful in certain fields, like the representationtheory of Sn.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 13: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/13.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
D1x21 =
∂
∂x1x2
1 − c∑i 6=1
(x1 − xi)−1(1− S1i)x2
1
=2x1 − c∑i 6=1
(x1 − xi)−1(x2
1 − x2i )
=2x1 − c∑i 6=1
(x1 + xi)
=2x1 − c(n − 1)x1 − c∑
xi
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 14: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/14.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
D1x21 =
∂
∂x1x2
1 − c∑i 6=1
(x1 − xi)−1(1− S1i)x2
1
=2x1 − c∑i 6=1
(x1 − xi)−1(x2
1 − x2i )
=2x1 − c∑i 6=1
(x1 + xi)
=2x1 − c(n − 1)x1 − c∑
xi
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 15: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/15.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
D1x21 =
∂
∂x1x2
1 − c∑i 6=1
(x1 − xi)−1(1− S1i)x2
1
=2x1 − c∑i 6=1
(x1 − xi)−1(x2
1 − x2i )
=2x1 − c∑i 6=1
(x1 + xi)
=2x1 − c(n − 1)x1 − c∑
xi
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 16: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/16.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
The set of linear transformations (i.e. matrices) from one (vector)space to itself, denoted End V is an algebra, with addition andscalar multiplication the obvious and multiplication beingcomposition.
A representation of an algebra is a V with a homomorphism (i.e.structure-preserving map) ρ to End V .
For any V , there is an automatic representation of End V with theidentity map, and for any A, with A an algebra, there is a obviousrepresentation of A with ρ(x)(y) = 0 for all x ∈ A, y ∈ V . Finally,if your algebra is a field, a representation is just a vector space
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 17: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/17.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
The set of linear transformations (i.e. matrices) from one (vector)space to itself, denoted End V is an algebra, with addition andscalar multiplication the obvious and multiplication beingcomposition.
A representation of an algebra is a V with a homomorphism (i.e.structure-preserving map) ρ to End V .
For any V , there is an automatic representation of End V with theidentity map, and for any A, with A an algebra, there is a obviousrepresentation of A with ρ(x)(y) = 0 for all x ∈ A, y ∈ V . Finally,if your algebra is a field, a representation is just a vector space
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 18: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/18.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
The set of linear transformations (i.e. matrices) from one (vector)space to itself, denoted End V is an algebra, with addition andscalar multiplication the obvious and multiplication beingcomposition.
A representation of an algebra is a V with a homomorphism (i.e.structure-preserving map) ρ to End V .
For any V , there is an automatic representation of End V with theidentity map, and for any A, with A an algebra, there is a obviousrepresentation of A with ρ(x)(y) = 0 for all x ∈ A, y ∈ V . Finally,if your algebra is a field, a representation is just a vector space
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 19: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/19.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Working in characteristic (mod) p > 0.
Cherednik algebras deform differential operators so that we use Diinstead of ∂
∂xi
In quantum mechanics, we can adjust the ∂∂xi
term in Dunkloperators by a factor of ~. Then, the xi are position vectors, Di aremomenta, and the extra part is accounting for Heisenberg’sUncertainty Principle.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 20: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/20.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Working in characteristic (mod) p > 0.
Cherednik algebras deform differential operators so that we use Diinstead of ∂
∂xi
In quantum mechanics, we can adjust the ∂∂xi
term in Dunkloperators by a factor of ~. Then, the xi are position vectors, Di aremomenta, and the extra part is accounting for Heisenberg’sUncertainty Principle.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 21: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/21.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Working in characteristic (mod) p > 0.
Cherednik algebras deform differential operators so that we use Diinstead of ∂
∂xi
In quantum mechanics, we can adjust the ∂∂xi
term in Dunkloperators by a factor of ~. Then, the xi are position vectors, Di aremomenta, and the extra part is accounting for Heisenberg’sUncertainty Principle.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 22: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/22.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
1
xi
xixj
xixjxkx1x2x3
xixjxkxlx2x6x7x8xix1x2x3
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 23: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/23.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Dunkl: “Singular Polynomials for the Symmetric Groups” provedresults concerning the kernel of Di − Dj .
Pavel et al.: “Representations of Rational Cherednik Algebras withMinimal Support and Torus Knots” discuss Cherednik algebraswith characteristic 0.
Devadas and Sun: “The Polynomial Representation of the TypeAn1 Rational Cherednik Algebra in Characteristic p|n” provessimilar kinds of results for characteristic p|n by extending certainresults from Pavel to positive characteristic.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 24: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/24.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Dunkl: “Singular Polynomials for the Symmetric Groups” provedresults concerning the kernel of Di − Dj .
Pavel et al.: “Representations of Rational Cherednik Algebras withMinimal Support and Torus Knots” discuss Cherednik algebraswith characteristic 0.
Devadas and Sun: “The Polynomial Representation of the TypeAn1 Rational Cherednik Algebra in Characteristic p|n” provessimilar kinds of results for characteristic p|n by extending certainresults from Pavel to positive characteristic.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 25: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/25.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
Dunkl: “Singular Polynomials for the Symmetric Groups” provedresults concerning the kernel of Di − Dj .
Pavel et al.: “Representations of Rational Cherednik Algebras withMinimal Support and Torus Knots” discuss Cherednik algebraswith characteristic 0.
Devadas and Sun: “The Polynomial Representation of the TypeAn1 Rational Cherednik Algebra in Characteristic p|n” provessimilar kinds of results for characteristic p|n by extending certainresults from Pavel to positive characteristic.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 26: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/26.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
We are trying the case p|n − 1, and will ideally develop techniquesor conjectures which will be used for p|n + 1 and ideally more. Wecurrently have three unproven conjectures
1. Devadas and Sun relied on a nice set of functions fi and thefact that n|Dfi so that the fi are singular in their case. Webelieve (n − 1)|Dfi fj .
2. We know that D2x2p+2i = 0 for small n ≡ 1 (mod p). We
hope to show that Dr x rp+ri = 0 whenever n ≡ r − 1 (mod p)
3. We also conjecture that, for t ≥ 3 (and showed this fort = 0, 1, 2):
Dy1tx s
1 =t∑
r=0(−c)r s!
(s − r)! f (r)
where f (0) =∑
a=s−t xa1 , f (1) =
∑a+b=s−t
∑i 6=1 xa
1 xbi ,
f (2) =∑
a+b+c=s−t∑
1 6=i 6=j 6=1 xa1 xb
i x cj , etc.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 27: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/27.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
We are trying the case p|n − 1, and will ideally develop techniquesor conjectures which will be used for p|n + 1 and ideally more. Wecurrently have three unproven conjectures
1. Devadas and Sun relied on a nice set of functions fi and thefact that n|Dfi so that the fi are singular in their case. Webelieve (n − 1)|Dfi fj .
2. We know that D2x2p+2i = 0 for small n ≡ 1 (mod p). We
hope to show that Dr x rp+ri = 0 whenever n ≡ r − 1 (mod p)
3. We also conjecture that, for t ≥ 3 (and showed this fort = 0, 1, 2):
Dy1tx s
1 =t∑
r=0(−c)r s!
(s − r)! f (r)
where f (0) =∑
a=s−t xa1 , f (1) =
∑a+b=s−t
∑i 6=1 xa
1 xbi ,
f (2) =∑
a+b+c=s−t∑
1 6=i 6=j 6=1 xa1 xb
i x cj , etc.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 28: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/28.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
We are trying the case p|n − 1, and will ideally develop techniquesor conjectures which will be used for p|n + 1 and ideally more. Wecurrently have three unproven conjectures
1. Devadas and Sun relied on a nice set of functions fi and thefact that n|Dfi so that the fi are singular in their case. Webelieve (n − 1)|Dfi fj .
2. We know that D2x2p+2i = 0 for small n ≡ 1 (mod p). We
hope to show that Dr x rp+ri = 0 whenever n ≡ r − 1 (mod p)
3. We also conjecture that, for t ≥ 3 (and showed this fort = 0, 1, 2):
Dy1tx s
1 =t∑
r=0(−c)r s!
(s − r)! f (r)
where f (0) =∑
a=s−t xa1 , f (1) =
∑a+b=s−t
∑i 6=1 xa
1 xbi ,
f (2) =∑
a+b+c=s−t∑
1 6=i 6=j 6=1 xa1 xb
i x cj , etc.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
![Page 29: Representations of Cherednik Algebras - MIT Mathematics€¦ · Representations of Cherednik Algebras Matthew Lipman Mentor: Gus Lonergan Sixth Annual MIT-PRIMES Conference May 21,](https://reader030.vdocument.in/reader030/viewer/2022040923/5e9e9f73fd4327494a21635b/html5/thumbnails/29.jpg)
Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
We are trying the case p|n − 1, and will ideally develop techniquesor conjectures which will be used for p|n + 1 and ideally more. Wecurrently have three unproven conjectures
1. Devadas and Sun relied on a nice set of functions fi and thefact that n|Dfi so that the fi are singular in their case. Webelieve (n − 1)|Dfi fj .
2. We know that D2x2p+2i = 0 for small n ≡ 1 (mod p). We
hope to show that Dr x rp+ri = 0 whenever n ≡ r − 1 (mod p)
3. We also conjecture that, for t ≥ 3 (and showed this fort = 0, 1, 2):
Dy1tx s
1 =t∑
r=0(−c)r s!
(s − r)! f (r)
where f (0) =∑
a=s−t xa1 , f (1) =
∑a+b=s−t
∑i 6=1 xa
1 xbi ,
f (2) =∑
a+b+c=s−t∑
1 6=i 6=j 6=1 xa1 xb
i x cj , etc.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras
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Intro A simple situation Dunkl Operators Representations The Algebra Diagram Previous Work Current Work Conclusions
I would like to thank:
My mentor, Gus Lonergan
My parents
and the PRIMES Program, especially Pavel for his help with theproject and Tanya for her help with the presentation.
Matthew Lipman Mentor: Gus LonerganCherednik Algebras