Smith Normal Form and Combinatorics Richard P . Stanley Smith - PowerPoint PPT Presentation

Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that det M t = 1 . 2 1 2 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 18

Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that det M t = 1 . 2 1 3 2 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 18

Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that det M t = 1 . 5 2 1 3 2 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 18

Determinantal algorithm Suppose all squares to the southeast of t have been filled. Insert into t the number n t so that det M t = 1 . 9 5 2 1 3 2 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 18

Uniqueness Easy to see: the numbers n t are well-defined and unique. Smith Normal Form and Combinatorics – p. 19

Uniqueness Easy to see: the numbers n t are well-defined and unique. Why? Expand det M t by the first row. The coefficient of n t is 1 by induction. Smith Normal Form and Combinatorics – p. 19

λ ( t ) If t ∈ λ , let λ ( t ) consist of all squares of λ to the southeast of t . Smith Normal Form and Combinatorics – p. 20

λ ( t ) If t ∈ λ , let λ ( t ) consist of all squares of λ to the southeast of t . λ = (4,4,3) t Smith Normal Form and Combinatorics – p. 20

λ ( t ) If t ∈ λ , let λ ( t ) consist of all squares of λ to the southeast of t . λ = (4,4,3) t λ t ( ) = (3,2) Smith Normal Form and Combinatorics – p. 20

u λ u λ = # { µ : µ ⊆ λ } Smith Normal Form and Combinatorics – p. 21

u λ u λ = # { µ : µ ⊆ λ } Example. u (2 , 1) = 5 : φ Smith Normal Form and Combinatorics – p. 21

u λ u λ = # { µ : µ ⊆ λ } Example. u (2 , 1) = 5 : φ There is a determinantal formula for u λ , due essentially to MacMahon and later Kreweras (not needed here). Smith Normal Form and Combinatorics – p. 21

Carlitz-Scoville-Roselle theorem Berlekamp (1963) first asked for n t (mod 2) in connection with a coding theory problem. Carlitz-Roselle-Scoville (1971): combinatorial interpretation of n t (over Z ). Smith Normal Form and Combinatorics – p. 22

Carlitz-Scoville-Roselle theorem Berlekamp (1963) first asked for n t (mod 2) in connection with a coding theory problem. Carlitz-Roselle-Scoville (1971): combinatorial interpretation of n t (over Z ). Theorem. n t = f ( λ ( t )) . Smith Normal Form and Combinatorics – p. 22

Carlitz-Scoville-Roselle theorem Berlekamp (1963) first asked for n t (mod 2) in connection with a coding theory problem. Carlitz-Roselle-Scoville (1971): combinatorial interpretation of n t (over Z ). Theorem. n t = f ( λ ( t )) . Proofs. 1. Induction (row and column operations). 2. Nonintersecting lattice paths. Smith Normal Form and Combinatorics – p. 22

An example 7 3 2 1 2 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 23

An example 7 3 2 1 2 1 1 1 1 1 φ Smith Normal Form and Combinatorics – p. 23

Many indeterminates For each square ( i, j ) ∈ λ , associate an indeterminate x ij (matrix coordinates). Smith Normal Form and Combinatorics – p. 24

Many indeterminates For each square ( i, j ) ∈ λ , associate an indeterminate x ij (matrix coordinates). x x x 11 12 13 x x 21 22 Smith Normal Form and Combinatorics – p. 24

A refinement of u λ � � u λ ( x ) = x ij µ ⊆ λ ( i,j ) ∈ λ/µ Smith Normal Form and Combinatorics – p. 25

A refinement of u λ � � u λ ( x ) = x ij µ ⊆ λ ( i,j ) ∈ λ/µ a b c c d e d e µ λ/µ λ � x ij = cde ( i,j ) ∈ λ/µ Smith Normal Form and Combinatorics – p. 25

An example a b c d e bce+ce+c abcde+bcde+bce+cde c+1 1 +e+1 +ce+de+c+e+1 de+e+1 e+1 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 26

A t � A t = x ij ( i,j ) ∈ λ ( t ) Smith Normal Form and Combinatorics – p. 27

A t � A t = x ij ( i,j ) ∈ λ ( t ) t a b c d e f g h i j k l m n o Smith Normal Form and Combinatorics – p. 27

A t � A t = x ij ( i,j ) ∈ λ ( t ) t a b c d e f g h i j k l m n o A t = bcdeghiklmo Smith Normal Form and Combinatorics – p. 27

The main theorem Theorem. Let t = ( i, j ) . Then M t has SNF diag( A ij , A i − 1 ,j − 1 , . . . , 1) . Smith Normal Form and Combinatorics – p. 28

The main theorem Theorem. Let t = ( i, j ) . Then M t has SNF diag( A ij , A i − 1 ,j − 1 , . . . , 1) . Proof. 1. Explicit row and column operations putting M t into SNF . 2. ( C. Bessenrodt ) Induction. Smith Normal Form and Combinatorics – p. 28

An example a b c d e bce+ce+c abcde+bcde+bce+cde c+1 1 +e+1 +ce+de+c+e+1 de+e+1 e+1 1 1 1 1 1 Smith Normal Form and Combinatorics – p. 29

An example a b c d e bce+ce+c abcde+bcde+bce+cde c+1 1 +e+1 +ce+de+c+e+1 de+e+1 e+1 1 1 1 1 1 SNF = diag( abcde, e, 1) Smith Normal Form and Combinatorics – p. 29

A special case Let λ be the staircase δ n = ( n − 1 , n − 2 , . . . , 1) . Set each x ij = q . Smith Normal Form and Combinatorics – p. 30

A special case Let λ be the staircase δ n = ( n − 1 , n − 2 , . . . , 1) . Set each x ij = q . Smith Normal Form and Combinatorics – p. 30

A special case Let λ be the staircase δ n = ( n − 1 , n − 2 , . . . , 1) . Set each x ij = q . � u δ n − 1 ( x ) x ij = q counts Dyck paths of length 2 n by � (scaled) area, and is thus the well-known q -analogue C n ( q ) of the Catalan number C n . Smith Normal Form and Combinatorics – p. 30

A q -Catalan example C 3 ( q ) = q 3 + q 2 + 2 q + 1 Smith Normal Form and Combinatorics – p. 31

A q -Catalan example C 3 ( q ) = q 3 + q 2 + 2 q + 1 � � C 4 ( q ) C 3 ( q ) 1 + q � � � � SNF ∼ diag( q 6 , q, 1) C 3 ( q ) 1 + q 1 � � � � � � 1 + q 1 1 � � Smith Normal Form and Combinatorics – p. 31

A q -Catalan example C 3 ( q ) = q 3 + q 2 + 2 q + 1 � � C 4 ( q ) C 3 ( q ) 1 + q � � � � SNF ∼ diag( q 6 , q, 1) C 3 ( q ) 1 + q 1 � � � � � � 1 + q 1 1 � � q -Catalan determinant previously known SNF is new Smith Normal Form and Combinatorics – p. 31

SNF of random matrices Huge literature on random matrices, mostly connected with eigenvalues. Very little work on SNF of random matrices over a PID. Smith Normal Form and Combinatorics – p. 32

Is the question interesting? Mat k ( n ) : all n × n Z -matrices with entries in [ − k, k ] (uniform distribution) p k ( n, d ) : probability that if M ∈ Mat k ( n ) and SNF ( M ) = ( e 1 , . . . , e n ) , then e 1 = d . Smith Normal Form and Combinatorics – p. 33

Is the question interesting? Mat k ( n ) : all n × n Z -matrices with entries in [ − k, k ] (uniform distribution) p k ( n, d ) : probability that if M ∈ Mat k ( n ) and SNF ( M ) = ( e 1 , . . . , e n ) , then e 1 = d . Recall: e 1 = gcd of 1 × 1 minors (entries) of M Smith Normal Form and Combinatorics – p. 33

Is the question interesting? Mat k ( n ) : all n × n Z -matrices with entries in [ − k, k ] (uniform distribution) p k ( n, d ) : probability that if M ∈ Mat k ( n ) and SNF ( M ) = ( e 1 , . . . , e n ) , then e 1 = d . Recall: e 1 = gcd of 1 × 1 minors (entries) of M Theorem. lim k →∞ p k ( n, d ) = 1 /d n 2 ζ ( n 2 ) Smith Normal Form and Combinatorics – p. 33

Work of Yinghui Wang Smith Normal Form and Combinatorics – p. 34

Work of Yinghui Wang ( ) Smith Normal Form and Combinatorics – p. 35

Work of Yinghui Wang ( ) Sample result. µ k ( n ) : probability that the SNF of a random A ∈ Mat k ( n ) satisfies e 1 = 2 , e 2 = 6 . µ ( n ) = lim k →∞ µ k ( n ) . Smith Normal Form and Combinatorics – p. 36

Conclusion   n ( n − 1) n 2 − 1 2 − i + � � µ ( n ) = 2 − n 2 2 − i  1 −  i =( n − 1) 2 i = n ( n − 1)+1 · 3 2 · 3 − ( n − 1) 2 (1 − 3 ( n − 1) 2 )(1 − 3 − n ) 2   n ( n − 1) n 2 − 1 p − i + � � �  . p − i ·  1 − p> 3 i =( n − 1) 2 i = n ( n − 1)+1 Smith Normal Form and Combinatorics – p. 37

A note on the proof uses a 2014 result of C. Feng , R. W. Nóbrega , F. R. Kschischang , and D. Silva , Communication over finite-chain-ring matrix channels: number of m × n matrices over Z /p s Z with specified SNF Smith Normal Form and Combinatorics – p. 38

A note on the proof uses a 2014 result of C. Feng , R. W. Nóbrega , F. R. Kschischang , and D. Silva , Communication over finite-chain-ring matrix channels: number of m × n matrices over Z /p s Z with specified SNF Note. Z /p s Z is not a PID, but SNF still exists because its ideals form a finite chain. Smith Normal Form and Combinatorics – p. 38

Cyclic cokernel κ ( n ) : probability that an n × n Z -matrix has SNF diag ( e 1 , e 2 , . . . , e n ) with e 1 = e 2 = · · · = e n − 1 = 1 . Smith Normal Form and Combinatorics – p. 39

Cyclic cokernel κ ( n ) : probability that an n × n Z -matrix has SNF diag ( e 1 , e 2 , . . . , e n ) with e 1 = e 2 = · · · = e n − 1 = 1 . � � 1 + 1 p 2 + 1 p 3 + · · · + 1 � p n p Theorem. κ ( n ) = ζ (2) ζ (3) · · · Smith Normal Form and Combinatorics – p. 39

Cyclic cokernel κ ( n ) : probability that an n × n Z -matrix has SNF diag ( e 1 , e 2 , . . . , e n ) with e 1 = e 2 = · · · = e n − 1 = 1 . � � 1 + 1 p 2 + 1 p 3 + · · · + 1 � p n p Theorem. κ ( n ) = ζ (2) ζ (3) · · · 1 Corollary. lim n →∞ κ ( n ) = ζ (6) � j ≥ 4 ζ ( j ) ≈ 0 . 846936 · · · . Smith Normal Form and Combinatorics – p. 39

Third example In collaboration with Tommy Wuxing Cai. Smith Normal Form and Combinatorics – p. 40

Third example In collaboration with . Smith Normal Form and Combinatorics – p. 40

Third example In collaboration with . Par( n ) : set of all partitions of n E.g., Par (4) = { 4 , 31 , 22 , 211 , 1111 } . Smith Normal Form and Combinatorics – p. 40

Third example In collaboration with . Par( n ) : set of all partitions of n E.g., Par (4) = { 4 , 31 , 22 , 211 , 1111 } . V n : real vector space with basis Par ( n ) Smith Normal Form and Combinatorics – p. 40

U Define U = U n : V n → V n +1 by � U ( λ ) = µ, µ where µ ∈ Par( n + 1) and µ i ≥ λ i ∀ i . Example. U (42211) = 52211 + 43211 + 42221 + 422111 Smith Normal Form and Combinatorics – p. 41

D Dually, define D = D n : V n → V n − 1 by � D ( λ ) = ν, ν where ν ∈ Par( n − 1) and ν i ≤ λ i ∀ i . Example. D (42211) = 32211 + 42111 + 4221 Smith Normal Form and Combinatorics – p. 42

Symmetric functions N OTE . Identify V n with the space Λ n Q of all homogeneous symmetric functions of degree n over Q , and identify λ ∈ V n with the Schur function s λ . Then U ( f ) = p 1 f, D ( f ) = ∂ f. ∂p 1 Smith Normal Form and Combinatorics – p. 43

Commutation relation Basic commutation relation: DU − UD = I Allows computation of eigenvalues of DU : V n → V n . ∂ Or note that the eigenvectors of ∂p 1 p 1 are the p λ ’s, λ ⊢ n . Smith Normal Form and Combinatorics – p. 44

Eigenvalues of DU Let p ( n ) = #Par( n ) = dim V n . Theorem. Let 1 ≤ i ≤ n + 1 , i � = n . Then i is an eigenvalue of D n +1 U n with multiplicity p ( n + 1 − i ) − p ( n − i ) . Hence n +1 � i p ( n +1 − i ) − p ( n − i ) . det D n +1 U n = i =1 Smith Normal Form and Combinatorics – p. 45

Eigenvalues of DU Let p ( n ) = #Par( n ) = dim V n . Theorem. Let 1 ≤ i ≤ n + 1 , i � = n . Then i is an eigenvalue of D n +1 U n with multiplicity p ( n + 1 − i ) − p ( n − i ) . Hence n +1 � i p ( n +1 − i ) − p ( n − i ) . det D n +1 U n = i =1 What about SNF of the matrix [ D n +1 U n ] (with respect to the basis Par( n ) )? Smith Normal Form and Combinatorics – p. 45

Conjecture of A. R. Miller, 2005 Conjecture (first form). Let e 1 , . . . , e p ( n ) be the eigenvalues of D n +1 U n . Then [ D n +1 U n ] has the same SNF as diag ( e 1 , . . . , e p ( n ) ) . Smith Normal Form and Combinatorics – p. 46

Conjecture of A. R. Miller, 2005 Conjecture (first form). Let e 1 , . . . , e p ( n ) be the eigenvalues of D n +1 U n . Then [ D n +1 U n ] has the same SNF as diag ( e 1 , . . . , e p ( n ) ) . Conjecture (second form). The diagonal entries of the SNF of [ D n +1 U n ] are: ( n + 1)( n − 1)! , with multiplicity 1 ( n − k )! with multiplicity p ( k + 1) − 2 p ( k ) + p ( k − 1) , 3 ≤ k ≤ n − 2 1, with multiplicity p ( n ) − p ( n − 1) + p ( n − 2) . Smith Normal Form and Combinatorics – p. 46

Not a trivial result N OTE . { p λ } λ ⊢ n is not an integral basis. Smith Normal Form and Combinatorics – p. 47

Another form m 1 ( λ ) : number of 1 ’s in λ M 1 ( n ) : multiset of all numbers m 1 ( λ ) + 1 , λ ∈ Par( n ) Let SNF of [ D n +1 U n ] be diag ( f 1 , f 2 , . . . , f p ( n ) ) . Conjecture (third form). f 1 is the product of the distinct entries of M 1 ( n ) ; f 2 is the product of the remaining distinct entries of M 1 ( n ) , etc. Smith Normal Form and Combinatorics – p. 48

An example: n = 6 Par(6) = { 6 , 51 , 42 , 33 , 411 , 321 , 222 , 3111 , 2211 , 21111 , 111111 } M 1 (6) = { 1 , 2 , 1 , 1 , 3 , 2 , 1 , 4 , 3 , 5 , 7 } ( f 1 , . . . , f 11 ) = (7 · 5 · 4 · 3 · 2 · 1 , 3 · 2 · 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1) = (840 , 6 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1) Smith Normal Form and Combinatorics – p. 49