Algebras in representations of the symmetric group St,when t is transcendental

Luke SciarappaMentor: Nathan Harman

5th Annual PRIMES ConferenceMay 16, 2015

Luke Sciarappa Algebras in representations of St PRIMES 2015 1 / 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

P. Deligne, 2004

Schur-Weyl duality between Sn and partition algebras CP∗(n)

CP∗(t) defined for any t ∈ CRep(St), though no St

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 ∈ CRep(St), though no St

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 ∈ CRep(St), though no St

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

Rep(St)

P. Deligne, 2004

Schur-Weyl duality between Sn and partition algebras CP∗(n)

CP∗(t) defined for any t ∈ CRep(St), though no St

Luke Sciarappa Algebras in representations of St PRIMES 2015 6 / 10

Simple Algebras in Rep(St)

Induction functor Rep(Sk) � Rep(St−k)→ Rep(St), interpolatingordinary induction

Inducing simple algebra from Rep(Sk) and trivial algebra fromRep(St−k) gives simple algebra in Rep(St)

Theorem

If t is transcendental, every simple algebra in Rep(St) is induced in thisway from a simple algebra in Rep(Sk) for some k.

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), interpolatingordinary induction

Inducing simple algebra from Rep(Sk) and trivial algebra fromRep(St−k) gives simple algebra in Rep(St)

Theorem

If t is transcendental, every simple algebra in Rep(St) is induced in thisway from a simple algebra in Rep(Sk) for some k.

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), interpolatingordinary induction

Inducing simple algebra from Rep(Sk) and trivial algebra fromRep(St−k) gives simple algebra in Rep(St)

Theorem

If t is transcendental, every simple algebra in Rep(St) is induced in thisway from a simple algebra in Rep(Sk) for some k.

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), interpolatingordinary induction

Inducing simple algebra from Rep(Sk) and trivial algebra fromRep(St−k) gives simple algebra in Rep(St)

Theorem

If t is transcendental, every simple algebra in Rep(St) is induced in thisway from a simple algebra in Rep(Sk) for some k.

Luke Sciarappa Algebras in representations of St PRIMES 2015 7 / 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

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

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

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 categorie des representations du groupe symetrique St, lorsque tn’est pas un entier naturel.In Proceedings of the International Colloquium on AlgebraicGroups 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 incomplex rank.J. of Algebra, 381, 2013.arXiv:1006.1381.

Luke Sciarappa Algebras in representations of St PRIMES 2015 10 / 10