Wreath Products in Algebraic Voting Theory

Transcription

1 Wreath Products in Algebraic Voting Theory Committed to Committees Ian Calaway 1 Joshua Csapo 2 Dr. Erin McNicholas 3 Eric Samelson 4 1 Macalester College 2 University of Michigan-Flint 3 Faculty Advisor Department of Mathematics Willamette University 4 Willamette University Willamette Mathematics Consortium REU, July 24, / 49

2 Outline 1 Algebraic Framework 2 Voting for Committees 3 Wreath Products 4 Decomposing a QS m [S n ]-module 2 / 49

3 Algebraic Framework Ballot Structure Vote Aggregation Point Allocation (Voting System) Election Results 3 / 49

4 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. 4 / 49

5 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. 4 / 49

6 Vote Aggregation: The Profile Space The components of the profile vector indicate the number of votes for a given tabloid option on the ballot. 5 / 49

7 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: 3 ABC 2 ACB p = 0 BAC 2 BCA 0 CAB 4 CBA 5 / 49

8 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. 6 / 49

9 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 QS n -modules. 6 / 49

10 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 QS n -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 ABC 0 ABC 2 ACB 2 ACB 0 BAC 2 (12) 3 BAC BCA 2 BCA 0 CAB 4 CAB 4 CBA 0 CBA 6 / 49

11 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 ABC 2 ACB 0 5 A BAC 2 2 B BCA 0 4 C CAB 4 CBA 7 / 49

12 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 ABC 2 ACB 0 10 A BAC 2 11 B BCA 0 12 C CAB 4 CBA 8 / 49

13 Voting Systems as Linear Transformations The previous Borda example can be represented as the following linear transformation: ABC ACB BAC BCA CAB CBA ( ) A T (2,1,0) = B C 9 / 49

14 Voting Systems as Linear Transformations The previous Borda example can be represented as the following linear transformation: ABC ACB BAC BCA CAB CBA ( ) A T (2,1,0) = B C Any system that gives candidates points based on their position within voters rankings can be represented by a similar linear transformation. 9 / 49

15 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 outcome receives for the election. T w ( p) = r 10 / 49

16 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 outcome receives for the election. T w ( p) = r The results spaces are also QS n -module. 10 / 49

17 Schur s Lemma Both profile spaces and results spaces are QS n -modules. These positional scoring systems are QS n -module homomorphisms, which can be proven with the previous fact and the fact that these systems are neutral. 11 / 49

18 Schur s Lemma Both profile spaces and results spaces are QS n -modules. These positional scoring systems are QS n -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. 11 / 49

19 Schur s Lemma Both profile spaces and results spaces are QS n -modules. These positional scoring systems are QS n -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 outcome of an election given a specific voting system. 11 / 49

20 Schur s Lemma Both profile spaces and results spaces are QS n -modules. These positional scoring systems are QS n -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 outcome of an election given a specific voting system. Definition A submodule is a subspace of a module that is invariant under the group action. 11 / 49

21 Key Points of Algebraic Voting Theory Domain 12 / 49

22 Key Points of Algebraic Voting Theory Domain Maps 12 / 49

23 Key Points of Algebraic Voting Theory Domain Maps Codomain 12 / 49

24 Voting for Committees What happens if we want to elect a subset of the candidates rather than a single candidate? 13 / 49

25 Voting for Committees What happens if we want to elect a subset of the candidates rather than a single candidate? 4 A 2 B 0 C 2 AB 1 AC 5 BC 3 ABC 13 / 49

26 Voting for Committees What happens if we want to elect a subset of the candidates rather than a single candidate? A B C AB AC BC ABC A B C AB AC BC ABC 13 / 49

27 Voting for Committees What happens if we want to elect a subset of the candidates rather than a single candidate? 4 A 2 B 0 C 1 AB 2 AB 1 AC 1 AC 1 BC 5 BC 3 ABC 13 / 49

28 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 / 49

29 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. 14 / 49

30 Wreath Products Let G be a group and S n be the symmetric group on the set N = {1, 2,..., n} of n elements. The wreath product of G by S n, denoted G[S n ], is the semidirect product G n S n where G[S n ] = {(f ; π) : f G n, π S n } 15 / 49

31 Wreath Products Let G be a group and S n be the symmetric group on the set N = {1, 2,..., n} of n elements. The wreath product of G by S n, denoted G[S n ], is the semidirect product G n S n where G[S n ] = {(f ; π) : f G n, π S n } In general, G n = {f : N G} 15 / 49

32 Wreath Product for G = S m In the case of G = S m, it is possible to express S m [S n ] in the following way: 16 / 49

33 Wreath Product for G = S m In the case of G = S m, it is possible to express S m [S n ] in the following way: G n = S m S m }{{} n times f = (σ 1,..., σ n ) where σ i S m 16 / 49

34 Wreath Product for G = S m In the case of G = S m, it is possible to express S m [S n ] in the following way: G n = S m S m }{{} n times f = (σ 1,..., σ n ) where σ i S m S m [S n ] = {(σ 1,..., σ n ; π) : σ i S m, π S n } 16 / 49

35 Wreath Products Some situations where wreath products are applicable: Basketball Team, S 3 [S 5 ] 15 players per team 5 positions, 3 players per position 2 Question 3 Response Referendum, S 3 [S 2 ] 2 proposals 3 possible answers: Yes, No, Obstain 17 / 49

36 Action of S m [S n ] Given the element structure (σ 1,..., σ n ; π) we can now concisely define multiplication in S m [S n ]: (σ 1, σ 2,..., σ n ; π)(δ 1, δ 2,..., δ n ; τ) = (σ 1 δ π 1 (1), σ 2 δ π 1 (2),..., σ n δ π 1 (n); πτ). for σ i, δ i S m and π, τ S n 18 / 49

37 Example: Selecting committees with S 2 [S 2 ] Suppose we have two departments A and B, each with two members a 1, a 2 and b 1, b 2, respectively. A B a 1 a 2 b 1 b 2 19 / 49

38 Example: Selecting committees with S 2 [S 2 ] Suppose we have two departments A and B, each with two members a 1, a 2 and b 1, b 2, respectively. A B a 1 a 2 b 1 b 2 There are 4 possible committees: W = {a 1, b 1 }, X = {a 1, b 2 }, Y = {a 2, b 1 } and Z = {a 2, b 2 } 19 / 49

39 Example: Selecting committes with S 2 [S 2 ] First, how does (σ 1, σ 2 ; π) S 2 [S 2 ] act on a single committee? π A B (σ 1, σ 2 ) a 1 a 2 b 1 b 2 20 / 49

40 Example: Selecting committees with S 2 [S 2 ] Consider the element ϕ = ((12), e; (12)) acting on committee W = {a 1, b 1 } (12) A B ((12), e) a 1 a 2 b 1 b 2 21 / 49

41 Example: Selecting committees with S 2 [S 2 ] ϕ = ((12), e; (12)) acting on committee W = {a 1, b 1 } A B a 1 (12) ϕ({a 1, b 1 }) ({a 2, b 1 }) a 2 b 1 e b 2 22 / 49

42 Example: Selecting committees with S 2 [S 2 ] ϕ = ((12), e; (12)) acting on committee W = {a 1, b 1 } A (12) B ϕ({a 1, b 1 }) ({a 2, b 1 }) a 1 a 2 b 1 b 2 23 / 49

43 Selecting committees with S 2 [S 2 ] ϕ = ((12), e; (12)) acting on committee W = {a 1, b 1 } A (12) B a 1 a 2 b 1 b 2 ϕ({a 1, b 1 }) ({a 2, b 1 }) ({a 1, b 2 }) W = {a 1, b 1 }, X = {a 1, b 2 }, Y = {a 2, b 1 } and Z = {a 2, b 2 } 24 / 49

44 Selecting committees with S 2 [S 2 ] ϕ = ((12), e; (12)) acting on committee W = {a 1, b 1 } A (12) B a 1 a 2 b 1 b 2 ϕ({a 1, b 1 }) ({a 2, b 1 }) ({a 1, b 2 }) W = {a 1, b 1 }, X = {a 1, b 2 }, Y = {a 2, b 1 } and Z = {a 2, b 2 } ϕ(w ) = X 24 / 49

45 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. 25 / 49

46 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. 1 a 1 b 1 1 W p = 0 a 1 b 2 3 = 0 X a 2 b 1 3 Y 2 a 2 b 2 2 Z 25 / 49

47 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 = a 1 b 1 a 1 b 2 a 2 b 1 a 2 b 2 = W X Y Z Let s apply ϕ = ((12), e; (12)) as before: ϕ( ) W X Y Z = X Z W Y 25 / 49

48 Group Action on the Profile and Results Space Another way of viewing the action of the wreath product: 1 a 1 b 1 0 a 1 b 2 3 a 2 b 1 2 a 2 b 2 26 / 49

49 Group Action on the Profile and Results Space Another way of viewing the action of the wreath product: 1 a 1 b 1 1 a 1 b 1 ϕ( 0 3 ) a 1 b 2 = ((12), e; (12)) 0 a 1 b 2 a 2 b 1 3 a 2 b 1 2 a 2 b 2 2 a 2 b 2 27 / 49

50 Group Action on the Profile and Results Space Another way of viewing the action of the wreath product: 1 a 1 b 1 1 a 1 b 1 ϕ( 0 3 ) a 1 b 2 = ((12), e); (12)) 0 a 1 b 2 a 2 b 1 3 a 2 b 1 2 a 2 b 2 2 a 2 b 2 28 / 49

51 Group Action on the Profile and Results Space Another way of viewing the action of the wreath product: 1 a 1 b 1 3 a 2 b 1 ϕ( 0 3 ) a 1 b 2 = ((12), e); (12)) 2 a 2 b 2 a 2 b 1 1 a 1 b 1 2 a 2 b 2 0 a 1 b 2 29 / 49

52 Another way of viewing the action of the wreath product: 1 a 1 b 1 0 a 2 b 1 ϕ( 0 3 ) a 1 b 2 = ((12), e); (12)) 2 a 2 b 2 a 2 b 1 1 a 1 b 1 2 a 2 b 2 3 a 1 b 2 30 / 49

53 Group Action on the Profile and Results Space Another way of viewing the action of the wreath product: 1 a 1 b 1 0 a 1 b 2 ϕ( 0 3 ) a 1 b 2 = ((12), e); (12)) 2 a 2 b 2 a 2 b 1 1 a 1 b 1 2 a 2 b 2 3 a 2 b 1 30 / 49

54 Group Action on the Profile and Results Space Another way of viewing the action of the wreath product: 1 a 1 b 1 0 a 1 b 2 ϕ( 0 3 ) a 1 b 2 = ((12), e); (12)) 2 a 2 b 2 a 2 b 1 1 a 1 b 1 2 a 2 b 2 3 a 2 b 1 1 W ϕ( 0 3 ) X Y 2 Z 0 = X Z Y W 30 / 49

55 QS m [S n ] Modules Defining the action of S m [S n ] on P and R allows us to view them as FG-modules; specifically Q(S m [S n ])-modules. In order to apply Schur s Lemma, all the remains is to show that a voting procedure T : P R is a QS m [S n ]-module homomorphism, then decompose each space into its respective simple submodules. 31 / 49

56 Decomposing a QS m [S n ]-module We will now outline the process of decomposing a QS m [S n ]-module into submodules. We will use algorithms by Rockmore (1995) and James and Liebeck (2001). Submodules of a QS m [S n ]-module are indexed by tuples of partitions which add up to n. S 3 [S 5 ] S (,, ) 32 / 49

57 Decomposing a QS m [S n ]-module We will now outline the process of decomposing a QS m [S n ]-module into submodules. We will use algorithms by Rockmore (1995) and James and Liebeck (2001). Submodules of a QS m [S n ]-module are indexed by tuples of partitions which add up to n. S 3 [S 5 ] S (,, ) 32 / 49

58 Decomposing a QS m [S n ]-module Find the h irreducible representations of S m Extend to irreducible reps of (S m) n Extend to irreducible reps of S m[s α] Induce representations of S m[s n] Find irreducible characters of S m[s n] Decompose QS m[s n]-module 33 / 49

59 Decomposing a QS m [S n ]-module Find the h irreducible representations of S m Extend to irreducible reps of (S m) n Extend to irreducible reps of S m[s α] Induce representations of S m[s n] 34 / 49

60 Decomposing a QS m [S n ]-module Find the h irreducible representations of S m Extend to irreducible reps of (S m) n Extend to irreducible reps of S m[s α] Indexed by weak compositions of n with h parts S 3[S 5] m = 3 = h, n = 5 α = (3, 0, 2) Induce representations of S m[s n] 34 / 49

61 Decomposing a QS m [S n ]-module Find the h irreducible representations of S m Extend to irreducible reps of (S m) n α = (3, 0, 2) Extend to irreducible reps of S m[s α] Indexed by h-tuples of partitions corresponding to α λ 1 = (,, ) Induce representations of S m[s n] 34 / 49

62 Decomposing a QS m [S n ]-module Find the h irreducible representations of S m Extend to irreducible reps of (S m) n α = (3, 0, 2) Extend to irreducible reps of S m[s α] Indexed by h-tuples of partitions corresponding to α λ 2 = (,, ) Induce representations of S m[s n] 34 / 49

63 Decomposing a QS m [S n ]-module Induce representations of S m[s n] Find irreducible characters of S m[s n] Decompose QS m[s n]-module 35 / 49

64 Decomposing a QS m [S n ]-module Induce representations of S m[s n] Find irreducible characters of S m[s n] Decompose QS m[s n]-module 1 Determine a basis v 1,..., v k for the QS m[s n]-module vector space. 2 For each irreducible character of S m[s n] calculate χ(g 1 )g v i g S m[s n] for each basis vector v i. The resulting vectors span the submodule S χ. 35 / 49

65 Example: The S 2 [S 2 ] case We will do the decomposition for the S 2 [S 2 ] case. Our QS 2 [S 2 ]-module is 4-dimensional and indexed by each of the possible pairs of candidates. Here is a basis: a 1 b 1 0 0, 1 0, 0 1, 0 a 1 b 2 0 a 2 b a 2 b 2 36 / 49

66 Example: The S 2 [S 2 ] case e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) (, ) (, ) (, ) (, ) (, ) / 49

67 Example: The S 2 [S 2 ] case e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) (, ) v 1 1 χ(g 1 )g = 0 0 (2e 2((12), (12); e) g S m[s n] a 1 b 1 = 0 0 e 0 a 1 b 2 0 ((12), (12); e) a 2 b a 2 b 2 38 / 49

68 Example: The S 2 [S 2 ] case e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) (, ) v 1 1 χ(g 1 )g = 0 0 (2e 2((12), (12); e) g S m[s n] a 1 b 1 = a 1 b 2 0 a 2 b a 2 b 2 39 / 49

69 Example: The S 2 [S 2 ] case e (e, (12); e) ((12), (12); e) (e, e; (12)) (e, (12); (12)) (, ) v 1 1 χ(g 1 )g = 0 0 (2e 2((12), (12); e) g S m[s n] a 1 b 1 = = 0 a 1 b 2 0 a 2 b a 2 b 2 39 / 49

70 Example: The S 2 [S 2 ] case We can now apply this same process to each of the other basis vectors v i of our space V, and we obtain the following submodule: S (, ) = , , 2 2, 0 0 = , / 49

71 Example: The S 2 [S 2 ] case 1 S (, ) = S (, ) = S (, ) = 0 S (, ) = 0 S (, ) = , / 49

72 Decompositions of S 2 [S n ] Conjecture (Lee, 2010 Thesis) For S 2 [S n ] with n 2, the results space decomposes into exactly λ S λ, the direct sum of irreducible submodules indexed by double trivial partitions λ = (λ 1, λ 2 ) (the flat partitions). 42 / 49

73 Example: The QS 3 [S 2 ] case As the wreath product S 3 [S 5 ] is not manageable by hand, we decided to decompose a corresponding QS 3 [S 2 ]-module instead and see if the flat partition property holds. S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) 43 / 49

74 Example: The QS 3 [S 2 ] case As the wreath product S 3 [S 5 ] is not manageable by hand, we decided to decompose a corresponding QS 3 [S 2 ]-module instead and see if the flat partition property holds. S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) 43 / 49

75 Example: The QS 3 [S 2 ] case As the wreath product S 3 [S 5 ] is not manageable by hand, we decided to decompose a corresponding QS 3 [S 2 ]-module instead and see if the flat partition property holds. S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) S (,, ) 43 / 49

76 Decompositions of S m [S n ] Conjecture (Calaway, Csapo, Samelson, 2015) For S m [S n ] 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). 44 / 49

77 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 S 3 [S 5 ]). Further investigate the decomposition of S 3 [S n ] and S m [S n ]. In particular, can we rederive/generalize a 2011 result of Caselli and Fulci? Characterize more fully the separability of simple submodules. 45 / 49

78 Acknowledgements Batman Erin McNicholas NSF grant which funded our research 46 / 49

79 The universe is an enormous direct product of representations of symmetry groups. -Steven Weinberg 47 / 49

80 Bibliography I [1] S. J. Brams and P. C. Fishburn. Approval voting. Birkhäuser, Boston, Mass., [2] F. Caselli and R. Fulci. Refined Gelfand models for wreath products. European J. Combin., 32(2): , [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 Amer. Math. Soc., Providence, RI, [4] Z. Daugherty. An algebraic approach to voting theory 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): , [6] P. C. Fishburn and A. Pekec. Approval voting for committees: Threshold approaches Available at pekec/publications/committeevotepekecfishburn.pdf. 48 / 49

81 Bibliography II [7] G. James and M. Liebeck. Representations and characters of groups. Cambridge University Press, New York, second edition, [8] S. C. Lee. Understanding voting for committees using wreath products Senior Thesis at Harvey Mudd College. [9] T. C. Ratliff. Some startling inconsistencies when electing committees. Soc. Choice Welf., 21(3): , [10] D. N. Rockmore. Fast Fourier transforms for wreath products. Appl. Comput. Harmon. Anal., 2(3): , [11] B. E. Sagan. The symmetric group, volume 203 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, Representations, combinatorial algorithms, and symmetric functions. 49 / 49