Extending the parking space Brendon Rhoades (joint with Andrew - PowerPoint PPT Presentation

Extending the parking space Brendon Rhoades (joint with Andrew Berget) UCSD

Parking Functions        A parking function of size n is a labeled Dyck path of size n : I a vertical run of size k is labeled with a subset of [ n ] of size k , I every letter in [ n ] appears once as a label. Defn: Park n = { parking functions of size n } . Fact: | Park n | = ( n + 1) n − 1 .

The Parking Space S n acts on Park n by label permutation .               Q: How does Park n decompose as an S n -module?

Vertical Run Partitions If D is a Dyck path of size n , get a vertical run partition λ ( D ) ` n . λ ( D ) = (3 , 2 , 2) ` 7

Coset Decomposition Given λ ` n , let S λ be the Young subgroup . M λ = S n / S λ = coset representation. Fact: Park n ⇠ D M λ ( D ) , where D ranges over all size n Dyck L = S n paths. Example: = S 3 M (3) � 3 M (2 , 1) � M (1 , 1 , 1) . Park 3 ⇠

Main Theorem Theorem: [Berget-R] There exists an S n +1 -module V n such that Res S n +1 ( V n ) ⇠ = S n Park n . S n

Main Theorem Theorem: [Berget-R] There exists an S n +1 -module V n such that Res S n +1 ( V n ) ⇠ = S n Park n . S n Riddle: Can you see the action of S n +1 on Park n ?

Main Theorem Theorem: [Berget-R] There exists an S n +1 -module V n such that Res S n +1 ( V n ) ⇠ = S n Park n . S n Riddle: Can you see the action of S n +1 on Park n ? Probably not.

Main Theorem Theorem: [Berget-R] There exists an S n +1 -module V n such that Res S n +1 ( V n ) ⇠ = S n Park n . S n Riddle: Can you see the action of S n +1 on Park n ? Probably not. Fact: Park n does not in general extend to S n +1 as a permutation module. Also, Park n does not in general extend to S n +2 at all.

Extendability of Representations Problem: Let M be an S n -module. Give a nice criterion for when M extends to S n +1 (or S n + r ).

Extendability of Representations Problem: Let M be an S n -module. Give a nice criterion for when M extends to S n +1 (or S n + r ). I The irrep S λ extends to S n +1 i ff λ ` n is a rectangle minus an outer corner.

Extendability of Representations Problem: Let M be an S n -module. Give a nice criterion for when M extends to S n +1 (or S n + r ). I The irrep S λ extends to S n +1 i ff λ ` n is a rectangle minus an outer corner. I The regular representation C [ S n ] extends to S n +2 . [Whitehouse]

Extendability of Representations Problem: Let M be an S n -module. Give a nice criterion for when M extends to S n +1 (or S n + r ). I The irrep S λ extends to S n +1 i ff λ ` n is a rectangle minus an outer corner. I The regular representation C [ S n ] extends to S n +2 . [Whitehouse] I The coset representation M λ does not extend to S 8 for λ = (3 , 2 , 2) ` 7.

Extendability of Representations Problem: Let M be an S n -module. Give a nice criterion for when M extends to S n +1 (or S n + r ). I The irrep S λ extends to S n +1 i ff λ ` n is a rectangle minus an outer corner. I The regular representation C [ S n ] extends to S n +2 . [Whitehouse] I The coset representation M λ does not extend to S 8 for λ = (3 , 2 , 2) ` 7. I The map Res : K 0 ( S n +1 ) ! K 0 ( S n ) is surjective over Q .

Graphs K n +1 = complete graph on [ n + 1]. � [ n +1] � A subgraph G ✓ is slim if the complement K n +1 � G is 2 connected.      

Polynomials � [ n +1] � To any subgraph G ✓ , we associate the polynomial 2 Y p ( G ) = ( x i � x j ) . ( i < j ) ∈ G       p ( G ) = ( x 2 � x 3 )( x 2 � x 6 )( x 3 � x 5 )( x 3 � x 6 )

Spaces Defn: Let V n ⇢ C [ x 1 , . . . , x n +1 ] be the subspace V n = span { p ( G ) : G ✓ K n +1 is slim } . Obs: V n is a graded S n +1 -module. Theorem: [Berget-R] Res S n +1 ( V n ) ⇠ = Park n . (Graded structure?) S n

Area Defn: The area of a Dyck path D is the number of boxes to the northwest of D . area( D ) = 11

Graded Main Result Theorem: [Berget-R] The S n -isomorphism type of the degree k piece V n ( k ) is M M λ ( D ) , D where D ranges over all size n Dyck paths with area k . Example: The graded S 3 -character of V 3 is q 0 M (3) + q 1 M (2 , 1) + 2 q 2 M (2 , 1) + q 3 M (1 , 1 , 1) .

Extended Structure � n � V n ( k ) = degree k piece of V n for k = 0 , 1 , . . . , . 2

Extended Structure � n � V n ( k ) = degree k piece of V n for k = 0 , 1 , . . . , . 2 Theorem: [Berget-R] V n ( k ) ⇠ = S n +1 Sym k ( V ) for 0  k < n , where V is the reflection representation of S n +1 .

Extended Structure � n � V n ( k ) = degree k piece of V n for k = 0 , 1 , . . . , . 2 Theorem: [Berget-R] V n ( k ) ⇠ = S n +1 Sym k ( V ) for 0  k < n , where V is the reflection representation of S n +1 . 2 π i n +1 . Theorem: [Berget-R] Let C = h (1 , 2 , . . . , n + 1) i and ζ = e Then ✓✓ n ◆◆ ⇠ = S n +1 Ind S n +1 V n (top) = V n ( ζ ) ⌦ sign . C 2

Open Problems Problem: Given a nice criterion for deciding whether an S n -module M extends to S n +1 (or S n + r ). Problem: Determine the full graded S n +1 -structure of V n . Problem: For n and k fixed, what is the maximum r so that V n ( k ) extends to S n + r ? I k = 0 ) r = 1 I k = 1 , n > 2 ) r = 1 I k = top ) r � 2.

Thanks for listening! A. Berget and B. Rhoades. Extending the parking space. arXiv: 1303.5505