Quantum Increasing Sequences generate Quantum Permutation Groups Pawe� l J´ oziak Faculty of Mathematics and Information Technology, Warsaw University of Technology 8 VIII 2019 Universitetet i Oslo Quantum groups and their analysis � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 1 / 21 n
Plan of the talk Quantum permutation groups and quantum increasing sequences 1 Motivations and the problem 2 The solution: � I + k , n � = S + 3 n � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 2 / 21 n
Plan of the talk Quantum permutation groups and quantum increasing sequences 1 Motivations and the problem 2 The solution: � I + k , n � = S + 3 n � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 3 / 21 n
Quantum permutation groups A bistochastic matrix over A is a square matrix u ∈ M n ⊗ A such that in each row and column the entries add up to 1 . n n � � u i j = 1 = u i j i =1 j =1 Definition The quantum permutation group over n -letter alphabet is a quantum n ) is the universal C ∗ of a n × n bistochastic group S + n such that C u ( S + matrix consisting of projections. This bistochastic matrix is a fundamental corepresentation: n � ∆( u i k ) = u i j ⊗ u j k j =1 � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 4 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences Fix k < n ∈ Z + . The set of length- k , [ n ] = { 1 , . . . , n } -valued increasing sequences is: � � I k , n = f : [ k ] → [ n ] : f ( i ) < f ( j ) whenever i < j Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . (1 , 2 , 3 , 5 , 8 , 7 , 4 , 6) Folklore/Example/Exercise Let b k , n : I k , n → S n be the above map. Then � b k , n ( I k , n ) � = S n . � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 5 / 21 n
Increasing sequences – matricial representation I k , n ∋ i = ( i 1 < . . . < i k ) �→ M ( i ) ∈ M n × k ( { 0 , 1 } ) � 1 if s = i t M ( i ) s , t = 0 otherwise Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 6 / 21 n
Increasing sequences – matricial representation I k , n ∋ i = ( i 1 < . . . < i k ) �→ M ( i ) ∈ M n × k ( { 0 , 1 } ) � 1 if s = i t M ( i ) s , t = 0 otherwise Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 6 / 21 n
Increasing sequences – matricial representation I k , n ∋ i = ( i 1 < . . . < i k ) �→ M ( i ) ∈ M n × k ( { 0 , 1 } ) � 1 if s = i t M ( i ) s , t = 0 otherwise Example Consider the sequence (2 < 3 < 5 < 6 < 8) ∈ I 5 , 8 . 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 6 / 21 n
Quantum increasing sequences Let k < n ∈ Z + and let C u ( I + k , n ) be the universal C ∗ -algebra generated by p i j , 1 ≤ i ≤ n , 1 ≤ j ≤ k subject to the following relations: 1 p i j p ∗ i j = p i j . 2 � n i =1 p i j = 1 for each 1 ≤ j ≤ k . 3 p i j p i ′ j ′ = 0 whenever j < j ′ and i ≥ i ′ . β k , n : C u ( S + n ) → C u ( I + k , n ) is given by: u i j �→ p i j for 1 ≤ i ≤ n , 1 ≤ j ≤ k , u i k + m �→ 0 for 1 ≤ m ≤ n − k and i < m or i > m + k , for 1 ≤ m ≤ n − k and 0 ≤ p ≤ k , m + p − 1 � u m + p k + m �→ p i p − p i +1 p +1 , i =0 where we set p 0 0 = 1 , p 0 i = p 0 i = p i k +1 = 0 for i ≥ 1. � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 7 / 21 n
Quantum increasing sequences Let k < n ∈ Z + and let C u ( I + k , n ) be the universal C ∗ -algebra generated by p i j , 1 ≤ i ≤ n , 1 ≤ j ≤ k subject to the following relations: 1 p i j p ∗ i j = p i j . 2 � n i =1 p i j = 1 for each 1 ≤ j ≤ k . 3 p i j p i ′ j ′ = 0 whenever j < j ′ and i ≥ i ′ . β k , n : C u ( S + n ) → C u ( I + k , n ) is given by: u i j �→ p i j for 1 ≤ i ≤ n , 1 ≤ j ≤ k , u i k + m �→ 0 for 1 ≤ m ≤ n − k and i < m or i > m + k , for 1 ≤ m ≤ n − k and 0 ≤ p ≤ k , m + p − 1 � u m + p k + m �→ p i p − p i +1 p +1 , i =0 where we set p 0 0 = 1 , p 0 i = p 0 i = p i k +1 = 0 for i ≥ 1. � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 7 / 21 n
Quantum increasing sequences Let k < n ∈ Z + and let C u ( I + k , n ) be the universal C ∗ -algebra generated by p i j , 1 ≤ i ≤ n , 1 ≤ j ≤ k subject to the following relations: 1 p i j p ∗ i j = p i j . 2 � n i =1 p i j = 1 for each 1 ≤ j ≤ k . 3 p i j p i ′ j ′ = 0 whenever j < j ′ and i ≥ i ′ . β k , n : C u ( S + n ) → C u ( I + k , n ) is given by: u i j �→ p i j for 1 ≤ i ≤ n , 1 ≤ j ≤ k , u i k + m �→ 0 for 1 ≤ m ≤ n − k and i < m or i > m + k , for 1 ≤ m ≤ n − k and 0 ≤ p ≤ k , m + p − 1 � u m + p k + m �→ p i p − p i +1 p +1 , i =0 where we set p 0 0 = 1 , p 0 i = p 0 i = p i k +1 = 0 for i ≥ 1. � I + k , n � = S + Pawe� l J´ oziak (MiNI PW) Oslo, 8 VIII 2019 7 / 21 n
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