Algebras in representations of the symmetric group St ,
when t is transcendental
Luke Sciarappa
Mentor: Nathan Harman
5th Annual PRIMES Conference
May 16, 2015
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
1 / 10
Symmetry: groups, actions, representations
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Symmetries of a cube
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Symmetries of a cube
Sn : ways to shuffle a pack of n cards / permutations of {1, . . . , n}
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Symmetries of a cube
Sn : ways to shuffle a pack of n cards / permutations of {1, . . . , n}
Action: group elements send objects to other objects
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Symmetries of a cube
Sn : ways to shuffle a pack of n cards / permutations of {1, . . . , n}
Action: group elements send objects to other objects
Examples
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Symmetries of a cube
Sn : ways to shuffle a pack of n cards / permutations of {1, . . . , n}
Action: group elements send objects to other objects
Examples
Symmetries of a cube act on vertices/edges/faces of cube, and on R3
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Symmetry: groups, actions, representations
Group: associative composition with identity and inverses
Examples
Symmetries of a cube
Sn : ways to shuffle a pack of n cards / permutations of {1, . . . , n}
Action: group elements send objects to other objects
Examples
Symmetries of a cube act on vertices/edges/faces of cube, and on R3
Representation: group acting linearly on a vector space
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
2 / 10
Categories
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
3 / 10
Categories
Category: mathematical objects; good notion of ‘the right maps’
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
3 / 10
Categories
Category: mathematical objects; good notion of ‘the right maps’
Examples
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
3 / 10
Categories
Category: mathematical objects; good notion of ‘the right maps’
Examples
sets; functions
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
3 / 10
Categories
Category: mathematical objects; good notion of ‘the right maps’
Examples
sets; functions
vector spaces; linear maps
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
3 / 10
Categories
Category: mathematical objects; good notion of ‘the right maps’
Examples
sets; functions
vector spaces; linear maps
G-representations; G-linear maps (linear maps ϕ s.t. ϕ(g · v) = g · ϕ(v))
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
3 / 10
Tensor categories
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Examples
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Examples
Vector spaces
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Examples
Vector spaces
Representations of a finite group
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Examples
Vector spaces
Representations of a finite group
In particular, for n ∈ N, representations of Sn =: Rep(Sn )
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Examples
Vector spaces
Representations of a finite group
In particular, for n ∈ N, representations of Sn =: Rep(Sn )
For t ∈ C, Rep(St ) (‘interpolation’ of Rep(Sn ))
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Tensor categories
Tensor category: category w/lots of extra structure
incl. tensor product: (X, Y ) 7→ X ⊗ Y ,
associative, commutative, unital (ACU)
Examples
Vector spaces
Representations of a finite group
In particular, for n ∈ N, representations of Sn =: Rep(Sn )
For t ∈ C, Rep(St ) (‘interpolation’ of Rep(Sn ))
Rep(St ) early example of tensor category besides reps of a group
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
4 / 10
Simple algebras
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Basic question of commutative algebra: what are simples?
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Basic question of commutative algebra: what are simples?
Simple algebra: algebra with no nontrivial quotients
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Basic question of commutative algebra: what are simples?
Simple algebra: algebra with no nontrivial quotients
Examples
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Basic question of commutative algebra: what are simples?
Simple algebra: algebra with no nontrivial quotients
Examples
Vector spaces — unique simple, C
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Basic question of commutative algebra: what are simples?
Simple algebra: algebra with no nontrivial quotients
Examples
Vector spaces — unique simple, C
Representations of a finite group — simples come from subgroups
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Simple algebras
Tensor category is good setting for commutative algebra
Algebra A has m : A ⊗ A → A and e : I → A; ACU
Basic question of commutative algebra: what are simples?
Simple algebra: algebra with no nontrivial quotients
Examples
Vector spaces — unique simple, C
Representations of a finite group — simples come from subgroups
Rep(St ) (‘interpolation’ of Rep(Sn )) — ???
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
5 / 10
Rep(St )
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
6 / 10
Rep(St )
P. Deligne, 2004
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
6 / 10
Rep(St )
P. Deligne, 2004
Schur-Weyl duality between Sn and partition algebras CP∗ (n)
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
6 / 10
Rep(St )
P. Deligne, 2004
Schur-Weyl duality between Sn and partition algebras CP∗ (n)
CP∗ (t) defined for any t ∈ C
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
6 / 10
Rep(St )
P. Deligne, 2004
Schur-Weyl duality between Sn and partition algebras CP∗ (n)
CP∗ (t) defined for any t ∈ C
Rep(St ), though no St
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
6 / 10
Simple Algebras in Rep(St )
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
7 / 10
Simple Algebras in Rep(St )
Induction functor Rep(Sk ) Rep(St−k ) → Rep(St ), interpolating
ordinary induction
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
7 / 10
Simple Algebras in Rep(St )
Induction functor Rep(Sk ) Rep(St−k ) → Rep(St ), interpolating
ordinary induction
Inducing simple algebra from Rep(Sk ) and trivial algebra from
Rep(St−k ) gives simple algebra in Rep(St )
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
7 / 10
Simple Algebras in Rep(St )
Induction functor Rep(Sk ) Rep(St−k ) → Rep(St ), interpolating
ordinary induction
Inducing simple algebra from Rep(Sk ) and trivial algebra from
Rep(St−k ) gives simple algebra in Rep(St )
Theorem
If t is transcendental, every simple algebra in Rep(St ) is induced in this
way from a simple algebra in Rep(Sk ) for some k.
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
7 / 10
Future directions
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
8 / 10
Future directions
Possible exotic algebras in algebraic t
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
8 / 10
Future directions
Possible exotic algebras in algebraic t
Noncommutative algebras, Lie algebras
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
8 / 10
Future directions
Possible exotic algebras in algebraic t
Noncommutative algebras, Lie algebras
Rep(GLt )?
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
8 / 10
Acknowledgments
Thanks to my mentor, Nathan Harman;
to Prof. Pavel Etingof, who suggested and supervised the project;
to the PRIMES program, for facilitating this research and conference;
to Dr. Tanya Khovanova, for advice on writing and presenting
mathematics;
and
to my parents, Pauline and Ken Sciarappa, for transport and support.
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
9 / 10
References
Jonathan Comes and Victor Ostrik.
On blocks of Deligne’s category Rep(St ).
Advances in Math., 226:1331–1377, 2011.
arXiv:0910.5695.
Pierre Deligne.
La catégorie des représentations du groupe symétrique St , lorsque t
n’est pas un entier naturel.
In Proceedings of the International Colloquium on Algebraic
Groups and Homogenous Spaces, January 2004.
Pavel Etingof.
Representation theory in complex rank, I.
Transformation Groups, 19(2):359–381, 2014.
arXiv:1401.6321.
Akhil Mathew.
Categories parametrized by schemes and representation theory in
complex rank.
J. of Algebra, 381, 2013.
arXiv:1006.1381.
Luke Sciarappa
Algebras in representations of St
PRIMES 2015
10 / 10