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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Products in Algebraic Voting Theory

Committed to Committees Ian Calaway1 Joshua Csapo2

• Dr. Erin McNicholas3

Eric Samelson4

1Macalester College 2University of Michigan-Flint 3Faculty Advisor

Department of Mathematics Willamette University

4Willamette University

Willamette Mathematics Consortium REU, July 24, 2015

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Outline

1

Algebraic Framework

2

Voting for Committees

3

Wreath Products

4

Decomposing a QSm[Sn]-module

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Algebraic Framework

Ballot Structure Vote Aggregation Point Allocation (Voting System) Election Results

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Ballots

Ballots → List of allowable tabloids A B C A B C A B C A B C These are elements of χ(3), χ(2,1), χ(1,2), and χ(1,1,1), respectively.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Ballots

Ballots → List of allowable tabloids A B C A B C A B C A B C These are elements of χ(3), χ(2,1), χ(1,2), and χ(1,1,1), respectively. Many elections focus on a single type of ballot structure. An exception to this is approval voting.

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Vote Aggregation: The Profile Space

The components of the profile vector indicate the number of votes for a given tabloid option on the ballot.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Vote Aggregation: The Profile Space

The components of the profile vector indicate the number of votes for a given tabloid option on the ballot. For an election in which voters give full rankings of three candidates (i.e. an election on the tabloids of χ(1,1,1)) the following is an example of a profile:

• p =

        3 2 2 4         ABC ACB BAC BCA CAB CBA

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Vote Aggregation: The Profile Space

The space of all possible profiles (i.e. all possible vote totals for a given ballot) is called the profile space.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Vote Aggregation: The Profile Space

The space of all possible profiles (i.e. all possible vote totals for a given ballot) is called the profile space. These profile spaces form QSn-modules.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Vote Aggregation: The Profile Space

The space of all possible profiles (i.e. all possible vote totals for a given ballot) is called the profile space. These profile spaces form QSn-modules. Definition An FG-module is a vector space over the field F where there is a representation of the group G on the vector space, and multiplication is understood as the associated group action.         3 2 2 4         ABC ACB BAC BCA CAB CBA → (12) →         2 3 2 4         ABC ACB BAC BCA CAB CBA

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Point Allocation: Voting Systems

Consider the following two voting systems for a full ranking (χ(1,1,1)) Plurality → (1, 0, 0)         3 2 2 4         ABC ACB BAC BCA CAB CBA →   5 2 4   A B C Borda → (2, 1, 0)

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Point Allocation: Voting Systems

Consider the following two voting systems for a full ranking (χ(1,1,1)) Plurality → (1, 0, 0) Borda → (2, 1, 0)         3 2 2 4         ABC ACB BAC BCA CAB CBA →   10 11 12   A B C

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Voting Systems as Linear Transformations

The previous Borda example can be represented as the following linear transformation: T(2,1,0) = ABC ACB BAC BCA CAB CBA

• 2

2 1 1 A 1 2 2 1 B 1 1 2 2 C

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Voting Systems as Linear Transformations

The previous Borda example can be represented as the following linear transformation: T(2,1,0) = ABC ACB BAC BCA CAB CBA

• 2

2 1 1 A 1 2 2 1 B 1 1 2 2 C Any system that gives candidates points based on their position within voters’ rankings can be represented by a similar linear transformation.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Results Space

When one of these voting systems is applied to a profile, a results vector is created. A results vector encodes the number of points that every potential

• utcome receives for the election.

Tw( p) = r

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Results Space

When one of these voting systems is applied to a profile, a results vector is created. A results vector encodes the number of points that every potential

• utcome receives for the election.

Tw( p) = r The results spaces are also QSn-module.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Schur’s Lemma

Both profile spaces and results spaces are QSn-modules. These positional scoring systems are QSn-module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Schur’s Lemma

Both profile spaces and results spaces are QSn-modules. These positional scoring systems are QSn-module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. Theorem Schur’s Lemma: Every nonzero module homomorphism between irreducible modules is an isomorphism.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Schur’s Lemma

Both profile spaces and results spaces are QSn-modules. These positional scoring systems are QSn-module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. Theorem Schur’s Lemma: Every nonzero module homomorphism between irreducible modules is an isomorphism. Schur’s Lemma helps us determine what actually affects the

• utcome of an election given a specific voting system.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Schur’s Lemma

Both profile spaces and results spaces are QSn-modules. These positional scoring systems are QSn-module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. Theorem Schur’s Lemma: Every nonzero module homomorphism between irreducible modules is an isomorphism. Schur’s Lemma helps us determine what actually affects the

• utcome of an election given a specific voting system.

Definition A submodule is a subspace of a module that is invariant under the group action.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Key Points of Algebraic Voting Theory

Domain

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Key Points of Algebraic Voting Theory

Domain → Maps

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Key Points of Algebraic Voting Theory

Domain → Maps → Codomain

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Voting for Committees

What happens if we want to elect a subset of the candidates rather than a single candidate?

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Voting for Committees

What happens if we want to elect a subset of the candidates rather than a single candidate?           4 2 2 1 5 3           A B C AB AC BC ABC →

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Voting for Committees

What happens if we want to elect a subset of the candidates rather than a single candidate?           4 2 2 1 5 3           A B C AB AC BC ABC →           1 1 1 1 1 1 1           A B C AB AC BC ABC

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Voting for Committees

What happens if we want to elect a subset of the candidates rather than a single candidate?           4 2 2 1 5 3           A B C AB AC BC ABC →   1 1 1   AB AC BC

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Basketball team

Consider a basketball team of 15 players where three people specialize in each of the 5 positions: Point Guard, Shooting Guard, Small Forward, Power Forward, and Center. Now we want to vote for the best team. Structural limitation on how people can vote Create maps for specific purposes Need something more than just permuting names of “candidates.”...

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Basketball team

Consider a basketball team of 15 players where three people specialize in each of the 5 positions: Point Guard, Shooting Guard, Small Forward, Power Forward, and Center. Now we want to vote for the best team. Structural limitation on how people can vote Create maps for specific purposes Need something more than just permuting names of “candidates.”... A wreath product is what we need.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Products

Let G be a group and Sn be the symmetric group on the set N = {1, 2, . . . , n} of n elements. The wreath product of G by Sn, denoted G[Sn], is the semidirect product G n ⋊ Sn where G[Sn] = {(f ; π) : f ∈ G n, π ∈ Sn}

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Products

Let G be a group and Sn be the symmetric group on the set N = {1, 2, . . . , n} of n elements. The wreath product of G by Sn, denoted G[Sn], is the semidirect product G n ⋊ Sn where G[Sn] = {(f ; π) : f ∈ G n, π ∈ Sn} In general, G n = {f : N → G}

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Product for G = Sm

In the case of G = Sm, it is possible to express Sm[Sn] in the following way:

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Product for G = Sm

In the case of G = Sm, it is possible to express Sm[Sn] in the following way: G n = Sm × · · · × Sm

• n times

f = (σ1, . . . , σn) where σi ∈ Sm

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Product for G = Sm

In the case of G = Sm, it is possible to express Sm[Sn] in the following way: G n = Sm × · · · × Sm

• n times

f = (σ1, . . . , σn) where σi ∈ Sm Sm[Sn] = {(σ1, . . . , σn; π) : σi ∈ Sm, π ∈ Sn}

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Wreath Products

Some situations where wreath products are applicable: Basketball Team, S3[S5] 15 players per team 5 positions, 3 players per position 2 Question 3 Response Referendum, S3[S2] 2 proposals 3 possible answers: Yes, No, Obstain

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Action of Sm[Sn]

Given the element structure (σ1, . . . , σn; π) we can now concisely define multiplication in Sm[Sn]: (σ1, σ2, . . . , σn; π)(δ1, δ2, . . . , δn; τ) = (σ1δπ−1(1), σ2δπ−1(2), . . . , σnδπ−1(n); πτ) for σi, δi ∈ Sm and π, τ ∈ Sn .

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: Selecting committees with S2[S2]

Suppose we have two departments A and B, each with two members a1, a2 and b1, b2, respectively. A

• B
• a1

a2 b1 b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: Selecting committees with S2[S2]

Suppose we have two departments A and B, each with two members a1, a2 and b1, b2, respectively. A

• B
• a1

a2 b1 b2 There are 4 possible committees: W = {a1, b1}, X = {a1, b2}, Y = {a2, b1} and Z = {a2, b2}

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: Selecting committes with S2[S2]

First, how does (σ1, σ2; π) ∈ S2[S2] act on a single committee? π

• (σ1, σ2)
• A
• B
• a1

a2 b1 b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: Selecting committees with S2[S2]

Consider the element ϕ = ((12), e; (12)) acting on committee W = {a1, b1} (12)

• ((12), e)
• A
• B
• a1

a2 b1 b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: Selecting committees with S2[S2]

ϕ = ((12), e; (12)) acting on committee W = {a1, b1} A

• B
• a1

(12)

a2

b1

e

• b2

ϕ({a1, b1}) ⇒ ({a2, b1})

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: Selecting committees with S2[S2]

ϕ = ((12), e; (12)) acting on committee W = {a1, b1} A

• (12) B
• a1

a2 b1 b2 ϕ({a1, b1}) ⇒ ({a2, b1})

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Selecting committees with S2[S2]

ϕ = ((12), e; (12)) acting on committee W = {a1, b1} A

• (12) B
• a1

a2 b1 b2 ϕ({a1, b1}) ⇒ ({a2, b1}) ⇒ ({a1, b2}) W = {a1, b1}, X = {a1, b2}, Y = {a2, b1} and Z = {a2, b2}

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Selecting committees with S2[S2]

ϕ = ((12), e; (12)) acting on committee W = {a1, b1} A

• (12) B
• a1

a2 b1 b2 ϕ({a1, b1}) ⇒ ({a2, b1}) ⇒ ({a1, b2}) W = {a1, b1}, X = {a1, b2}, Y = {a2, b1} and Z = {a2, b2} ϕ(W ) = X

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

We can now define the group action of the wreath product on the profile and results space, P and R, respectively.

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Group Action on the Profile and Results Space

We can now define the group action of the wreath product on the profile and results space, P and R, respectively. Consider an element p ∈ P.

• p =

    1 3 2     a1b1 a1b2 a2b1 a2b2 =     1 3 2     W X Y Z

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

We can now define the group action of the wreath product on the profile and results space, P and R, respectively. Consider an element p ∈ P.

• p =

    1 3 2     a1b1 a1b2 a2b1 a2b2 =     1 3 2     W X Y Z Let’s apply ϕ = ((12), e; (12)) as before: ϕ(     1 3 2    ) W X Y Z =     2 1 3     X Z W Y

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

Another way of viewing the action of the wreath product:     1 3 2     a1b1 a1b2 a2b1 a2b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

Another way of viewing the action of the wreath product: ϕ(     1 3 2    ) a1b1 a1b2 a2b1 a2b2 = ((12), e; (12))     1 3 2     a1b1 a1b2 a2b1 a2b2

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Group Action on the Profile and Results Space

Another way of viewing the action of the wreath product: ϕ(     1 3 2    ) a1b1 a1b2 a2b1 a2b2 = ((12), e); (12))     1 3 2     a1 b1 a1 b2 a2 b1 a2 b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

Another way of viewing the action of the wreath product: ϕ(     1 3 2    ) a1b1 a1b2 a2b1 a2b2 = ((12), e); (12))     3 2 1     a2 b1 a2 b2 a1 b1 a1 b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Another way of viewing the action of the wreath product: ϕ(     1 3 2    ) a1b1 a1b2 a2b1 a2b2 = ((12), e); (12))     2 1 3     a2b1 a2b2 a1b1 a1b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

Another way of viewing the action of the wreath product: ϕ(     1 3 2    ) a1b1 a1b2 a2b1 a2b2 = ((12), e); (12))     2 1 3     a1b2 a2b2 a1b1 a2b1

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Group Action on the Profile and Results Space

Another way of viewing the action of the wreath product: ϕ(     1 3 2    ) a1b1 a1b2 a2b1 a2b2 = ((12), e); (12))     2 1 3     a1b2 a2b2 a1b1 a2b1 ϕ(     1 3 2    ) W X Y Z =     2 1 3     X Z Y W

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

QSm[Sn] Modules

Defining the action of Sm[Sn] on P and R allows us to view them as FG-modules; specifically Q(Sm[Sn])-modules. In order to apply Schur’s Lemma, all the remains is to show that a voting procedure T : P → R is a QSm[Sn]-module homomorphism, then decompose each space into its respective simple submodules.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

We will now outline the process of decomposing a QSm[Sn]-module into submodules. We will use algorithms by Rockmore (1995) and James and Liebeck (2001). Submodules of a QSm[Sn]-module are indexed by tuples of partitions which add up to n. S3[S5] S(

, , )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

We will now outline the process of decomposing a QSm[Sn]-module into submodules. We will use algorithms by Rockmore (1995) and James and Liebeck (2001). Submodules of a QSm[Sn]-module are indexed by tuples of partitions which add up to n. S3[S5] S(

,∅, )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Find the h irreducible representations of Sm Extend to irreducible reps of (Sm)n Extend to irreducible reps of Sm[Sα] Induce representations of Sm[Sn] Find irreducible characters of Sm[Sn] Decompose QSm[Sn]-module

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Find the h irreducible representations of Sm Extend to irreducible reps of (Sm)n Extend to irreducible reps of Sm[Sα] Induce representations of Sm[Sn]

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Find the h irreducible representations of Sm Extend to irreducible reps of (Sm)n Extend to irreducible reps of Sm[Sα] Induce representations of Sm[Sn] Indexed by weak compositions of n with h parts S3[S5] m = 3 = h, n = 5 α = (3, 0, 2)

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Find the h irreducible representations of Sm Extend to irreducible reps of (Sm)n Extend to irreducible reps of Sm[Sα] Induce representations of Sm[Sn] α = (3, 0, 2) Indexed by h-tuples of partitions corresponding to α λ1 = ( , ∅, )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Find the h irreducible representations of Sm Extend to irreducible reps of (Sm)n Extend to irreducible reps of Sm[Sα] Induce representations of Sm[Sn] α = (3, 0, 2) Indexed by h-tuples of partitions corresponding to α λ2 = ( , ∅, )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Induce representations of Sm[Sn] Find irreducible characters of Sm[Sn] Decompose QSm[Sn]-module

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decomposing a QSm[Sn]-module

Induce representations of Sm[Sn] Find irreducible characters of Sm[Sn] Decompose QSm[Sn]-module

1

Determine a basis v1, . . . , vk for the QSm[Sn]-module vector space.

2

For each irreducible character of Sm[Sn] calculate vi  

• g∈Sm[Sn]

χ(g −1)g   for each basis vector vi. The resulting vectors span the submodule Sχ.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

We will do the decomposition for the S2[S2] case. Our QS2[S2]-module is 4-dimensional and indexed by each of the possible pairs of candidates. Here is a basis:     1     ,     1     ,     1     ,     1     a1b1 a1b2 a2b1 a2b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) ( , ∅) 1 1 1 1 1 ( , ∅) 1 1 1 −1 −1 ( , ) 2 −2 (∅, ) 1 −1 1 1 −1 (∅, ) 1 −1 1 −1 1

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) ( , ) 2 −2

v1  

• g∈Sm[Sn]

χ(g−1)g   =     1     (2e − 2((12), (12); e) =     2     e −     2     a1b1 a1b2 a2b1 a2b2 ((12), (12); e)

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) ( , ) 2 −2

v1  

• g∈Sm[Sn]

χ(g−1)g   =     1     (2e − 2((12), (12); e) =     2     −     2     a1b1 a1b2 a2b1 a2b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) ( , ) 2 −2

v1  

• g∈Sm[Sn]

χ(g−1)g   =     1     (2e − 2((12), (12); e) =     2     −     2     =     2 −2     a1b1 a1b2 a2b1 a2b2

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

We can now apply this same process to each of the other basis vectors vi of our space V , and we obtain the following submodule: S(

, ) =

    2 −2     ,     2 −2     ,     −2 2     ,     −2 2     =     1 −1     ,     1 −1    

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The S2[S2] case

S(

,∅) =

    1 1 1 1     S(

,∅) =

S(∅,

) =

    1 −1 −1 1     S(∅,

) =

S(

, ) =

    1 −1     ,     1 −1    

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decompositions of S2[Sn]

Conjecture (Lee, 2010 Thesis) For S2[Sn] with n ≥ 2, the results space decomposes into exactly

• λ Sλ, the direct sum of irreducible submodules indexed by

double trivial partitions λ = (λ1, λ2) (the “flat” partitions).

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The QS3[S2] case

As the wreath product S3[S5] is not manageable by hand, we decided to decompose a corresponding QS3[S2]-module instead and see if the “flat partition” property holds. S(

,∅,∅)

S(

,∅,∅)

S(

, ,∅)

S(∅,

,∅)

S(∅,

,∅)

S(∅,

, )

S(∅,∅,

)

S(∅,∅,

)

S(

,∅, )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The QS3[S2] case

As the wreath product S3[S5] is not manageable by hand, we decided to decompose a corresponding QS3[S2]-module instead and see if the “flat partition” property holds. S(

,∅,∅)

S(

,∅,∅)

S(

, ,∅)

S(∅,

,∅)

S(∅,

,∅)

S(∅,

, )

S(∅,∅,

)

S(∅,∅,

)

S(

,∅, )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Example: The QS3[S2] case

As the wreath product S3[S5] is not manageable by hand, we decided to decompose a corresponding QS3[S2]-module instead and see if the “flat partition” property holds. S(

,∅,∅)

S(

,∅,∅)

S(

, ,∅)

S(∅,

,∅)

S(∅,

,∅)

S(∅,

, )

S(∅,∅,

)

S(∅,∅,

)

S(

,∅, )

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Decompositions of Sm[Sn]

Conjecture (Calaway, Csapo, Samelson, 2015) For Sm[Sn] with m, n ≥ 2, the results space decomposes into a direct sum composed only of irreducible submodules indexed by h-tuple trivial partitions (the “flat” partitions).

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Going Forward

Now that we understand how to decompose these spaces, apply these tools to the voting schemes we discussed previously (especially in the case of S3[S5]). Further investigate the decomposition of S3[Sn] and Sm[Sn]. In particular, can we rederive/generalize a 2011 result of Caselli and Fulci? Characterize more fully the separability of simple submodules.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Acknowledgements

Batman Erin McNicholas NSF grant which funded our research

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

The universe is an enormous direct product

• f representations of symmetry groups.
• Steven Weinberg

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Bibliography I

[1] S. J. Brams and P. C. Fishburn. Approval voting. Birkh¨ auser, Boston, Mass., 1983. [2] F. Caselli and R. Fulci. Refined Gelfand models for wreath products. European J. Combin., 32(2):198–216, 2011. [3] K.-D. Crisman. The Borda count, the Kemeny rule, and the

• permutahedron. In The mathematics of decisions, elections, and games,

volume 624 of Contemp. Math., pages 101–134. Amer. Math. Soc., Providence, RI, 2014. [4] Z. Daugherty. An algebraic approach to voting theory. 2005. Senior Thesis at Harvey Mudd College. [5] Z. Daugherty, A. K. Eustis, G. Minton, and M. E. Orrison. Voting, the symmetric group, and representation theory. Amer. Math. Monthly, 116(8):667–687, 2009. [6] P. C. Fishburn and A. Pekec. Approval voting for committees: Threshold

• approaches. 2004. Available at

http://people.duke.edu/ pekec/Publications/CommitteeVotePekecFishburn.pdf.

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Algebraic Framework Voting for Committees Wreath Products Decomposing a QSm[Sn]-module References

Bibliography II

[7] G. James and M. Liebeck. Representations and characters of groups. Cambridge University Press, New York, second edition, 2001. [8] S. C. Lee. Understanding voting for committees using wreath products.

• 2010. Senior Thesis at Harvey Mudd College.

[9] T. C. Ratliff. Some startling inconsistencies when electing committees.

• Soc. Choice Welf., 21(3):433–454, 2003.

[10] D. N. Rockmore. Fast Fourier transforms for wreath products. Appl.

• Comput. Harmon. Anal., 2(3):279–292, 1995.

[11] B. E. Sagan. The symmetric group, volume 203 of Graduate Texts in

• Mathematics. Springer-Verlag, New York, second edition, 2001.

Representations, combinatorial algorithms, and symmetric functions.

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