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1 Smith normal forms of matrices associated with the Grassmann graphs of lines in PG(n, q) Peter Sin, U. of Florida UD Discrete Math Seminar November th, 206

2 Smith normal forms Graphs from lines Smith normal forms Chip-firing game Cross-characteristics Defining characteristic

3 Critical groups of graphs Graphs from lines Smith normal forms Chip-firing game Cross-characteristics Defining characteristic

4 Grassmann and skew lines graphs q = p t, p prime, V = F n q,

5 Grassmann and skew lines graphs q = p t, p prime, V = F n q, Γ = Γ (n, q) Grassman graph, vertices are lines of PG(V ), i.e. 2-diml subspaces of V. Two vertices lie on an edge iff the subspaces are distinct and have nonzero intersection.

6 Grassmann and skew lines graphs q = p t, p prime, V = F n q, Γ = Γ (n, q) Grassman graph, vertices are lines of PG(V ), i.e. 2-diml subspaces of V. Two vertices lie on an edge iff the subspaces are distinct and have nonzero intersection. Γ the complementary graph, is the skew lines graph.

7 Strongly Regular Graphs Definition A strongly regular graph with parameters (v, k, λ, µ) is a k-regular graph such that any two adjacent vertices have λ neighbors in common; and any two nonadjacent vertices have µ neighbors in common. The adjacency matrix of a SRG has three eigenvalues k, r, s.

8 Strongly Regular Graphs Definition A strongly regular graph with parameters (v, k, λ, µ) is a k-regular graph such that any two adjacent vertices have λ neighbors in common; and any two nonadjacent vertices have µ neighbors in common. The adjacency matrix of a SRG has three eigenvalues k, r, s. Γ is a SRG v = [ n 2 ] q, k = q(q + )[ ] n 2 q, λ = [ ] n q + q2 2, µ = (q + ) 2, r = [ n ] q and s = (q + ) [ n ] q.

9 Strongly Regular Graphs Definition A strongly regular graph with parameters (v, k, λ, µ) is a k-regular graph such that any two adjacent vertices have λ neighbors in common; and any two nonadjacent vertices have µ neighbors in common. The adjacency matrix of a SRG has three eigenvalues k, r, s. Γ is a SRG v = [ n 2 ] q, k = q(q + )[ ] n 2 q, λ = [ ] n q + q2 2, µ = (q + ) 2, r = [ n ] q and s = (q + ) [ ] n q. Γ is also a SRG. So we have two families of SRGs paramtrized by n, p and t.

10 Critical groups of graphs Graphs from lines Smith normal forms Chip-firing game Cross-characteristics Defining characteristic

11 Smith Normal form A(G), an adjacency matrix of a graph G.

12 Smith Normal form A(G), an adjacency matrix of a graph G. L(G) = D(G) A(G), Laplacian matrix.

13 Smith Normal form A(G), an adjacency matrix of a graph G. L(G) = D(G) A(G), Laplacian matrix. The Smith normal forms of A(G) and L(G) are invariants of G.

14 Smith Normal form A(G), an adjacency matrix of a graph G. L(G) = D(G) A(G), Laplacian matrix. The Smith normal forms of A(G) and L(G) are invariants of G. S(G), the abelian group whose cyclic decomposition is given by the SNF of A(G), (Smith group).

15 Smith Normal form A(G), an adjacency matrix of a graph G. L(G) = D(G) A(G), Laplacian matrix. The Smith normal forms of A(G) and L(G) are invariants of G. S(G), the abelian group whose cyclic decomposition is given by the SNF of A(G), (Smith group). K (G), the finite part of the abelian group whose cyclic decomposition is given by the SNF of L(G) (critical group, sandpile group, jacobian).

16 Smith Normal form A(G), an adjacency matrix of a graph G. L(G) = D(G) A(G), Laplacian matrix. The Smith normal forms of A(G) and L(G) are invariants of G. S(G), the abelian group whose cyclic decomposition is given by the SNF of A(G), (Smith group). K (G), the finite part of the abelian group whose cyclic decomposition is given by the SNF of L(G) (critical group, sandpile group, jacobian). K (G) = number of spanning trees (Kirchhoff s Matrix-tree Theorem).

17 Smith Normal form A(G), an adjacency matrix of a graph G. L(G) = D(G) A(G), Laplacian matrix. The Smith normal forms of A(G) and L(G) are invariants of G. S(G), the abelian group whose cyclic decomposition is given by the SNF of A(G), (Smith group). K (G), the finite part of the abelian group whose cyclic decomposition is given by the SNF of L(G) (critical group, sandpile group, jacobian). K (G) = number of spanning trees (Kirchhoff s Matrix-tree Theorem). Survey article on SNFs in combinatorics by R. Stanley (JCTA 206).

18 Critical groups of graphs Graphs from lines Smith normal forms Chip-firing game Cross-characteristics Defining characteristic

19 Rules A configuration on the graph G is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex.

20 Rules A configuration on the graph G is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips.

21 Rules A configuration on the graph G is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips.

22 Rules A configuration on the graph G is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips.

23 Rules A configuration on the graph G is an assignment of a nonnegative integer s(v) to each round vertex v and v s(v) to the square vertex. A round vertex v can be fired if it has at least deg(v) chips. The square vertex is fired only when no others can be fired.

24 Example game -2 0

25 Example game

26 Example game

27 Example game

28 Example game

29 Example game

30 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s.

31 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = s(v) x(v) deg(v) + (v,w) E x(w)

32 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = s(v) x(v) deg(v) + (v,w) E x(w) s = s L(G)x.

33 Relation with Laplacian Start with a configuration s and fire vertices in a sequence where each vertex v is fired x(v) times, ending up with configuration s. s (v) = s(v) x(v) deg(v) + (v,w) E x(w) s = s L(G)x. A configuration is stable if no round vertex can be fired, recurrent if there is a sequence of firings leading back to the same configuration, critical if recurrent and stable. Theorem (Dhar, Björner-Lovász, Biggs, Gabrielov,...) Consider the chip-firing game on a connected graph G. Any starting configuration leads to a unique critical configuration. The set of critical configurations has a natural group operation making it isomorphic to the critical group K (G).

34 GL(n, q)-permutation modules We (Ducey-S) compute the SNF for A = A(Γ(n, q)), L = L(Γ(n, q)), A = A(Γ (n, q)), L = L(Γ (n, q)).

35 GL(n, q)-permutation modules We (Ducey-S) compute the SNF for A = A(Γ(n, q)), L = L(Γ(n, q)), A = A(Γ (n, q)), L = L(Γ (n, q)). L r, the set of r-diml. subspaces of V.

36 GL(n, q)-permutation modules We (Ducey-S) compute the SNF for A = A(Γ(n, q)), L = L(Γ(n, q)), A = A(Γ (n, q)), L = L(Γ (n, q)). L r, the set of r-diml. subspaces of V. A and L define Z GL(n, q)-module homomorphisms A, L : Z L 2 Z L 2

37 GL(n, q)-permutation modules We (Ducey-S) compute the SNF for A = A(Γ(n, q)), L = L(Γ(n, q)), A = A(Γ (n, q)), L = L(Γ (n, q)). L r, the set of r-diml. subspaces of V. A and L define Z GL(n, q)-module homomorphisms A, L : Z L 2 Z L 2 SNF can be computed one prime at a time (elementary divisors).

38 GL(n, q)-permutation modules We (Ducey-S) compute the SNF for A = A(Γ(n, q)), L = L(Γ(n, q)), A = A(Γ (n, q)), L = L(Γ (n, q)). L r, the set of r-diml. subspaces of V. A and L define Z GL(n, q)-module homomorphisms A, L : Z L 2 Z L 2 SNF can be computed one prime at a time (elementary divisors). Fix prime l M = Z L 2 l, M = FL 2 l, α {A, L}.

39 GL(n, q)-permutation modules We (Ducey-S) compute the SNF for A = A(Γ(n, q)), L = L(Γ(n, q)), A = A(Γ (n, q)), L = L(Γ (n, q)). L r, the set of r-diml. subspaces of V. A and L define Z GL(n, q)-module homomorphisms A, L : Z L 2 Z L 2 SNF can be computed one prime at a time (elementary divisors). Fix prime l M = Z L 2 l, M = FL 2 l, α {A, L}. M = M 0 M M r = Ker(α) 0. M = M 0 M M r = Ker(α) 0.

40 e i = e i (α) := multiplicity of l i as an elementary divisor of α. (e 0 = rank(α)).

41 e i = e i (α) := multiplicity of l i as an elementary divisor of α. (e 0 = rank(α)). dim M a = + i a e i.

42 e i = e i (α) := multiplicity of l i as an elementary divisor of α. (e 0 = rank(α)). dim M a = + i a e i. All quotients M a /M a+ are F l GL(n, q)-modules, so the number of nonzero e i is at most the composition length of M as a F l GL(n, q)-module.

43 Critical groups of graphs Graphs from lines Smith normal forms Chip-firing game Cross-characteristics Defining characteristic

44 l-modular representations of GL(n, q) l p

45 l-modular representations of GL(n, q) l p Theory due to G. D. James.

46 l-modular representations of GL(n, q) l p Theory due to G. D. James. M is a q-analogue of the F l -permutation module of subsets of size 2 in a set of size n.

47 l-modular representations of GL(n, q) l p Theory due to G. D. James. M is a q-analogue of the F l -permutation module of subsets of size 2 in a set of size n. M has 3 isomorphism types of composition factors.

48 l-modular representations of GL(n, q) l p Theory due to G. D. James. M is a q-analogue of the F l -permutation module of subsets of size 2 in a set of size n. M has 3 isomorphism types of composition factors. The composition length is 6 and does not grow with n or t or p, but depends on certain divisibility conditions.

49 l-modular representations of GL(n, q) l p Theory due to G. D. James. M is a q-analogue of the F l -permutation module of subsets of size 2 in a set of size n. M has 3 isomorphism types of composition factors. The composition length is 6 and does not grow with n or t or p, but depends on certain divisibility conditions. Based on James results, it is easy to work out the submodule structure of M in all cases.

50 l [ ] n 2 q l [ ] n 2 q l q + l q + [ ] l n l [ ] n q q M = F l D D 2 M = F l D D 2 M = D F l D 2 F l N/A l n 2 M = F l D D 2 N/A F l l n 2 M = D D 2 F l Table : l ( ) n q N/A D

51 [ l q+ ] l n q l [ ] n 2 q F l M= D D 2 F l l q+ l n 2 N/A l [ n 2 ] q D l [ ] n 2 q N/A l [ n 2 ] q D l q+ l n M = F l D 2 F l D 2 N/A M=F l Table : l ( n ) q D F l D 2 F l D M = F l D 2 F l D

52 Example: l q +, n even Assume l q +, n even.

53 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2.

54 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2. r has multiplicity f = [ n ] q, s has multiplicity g = [ n 2 ] q [ n ] q.

55 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2. r has multiplicity f = [ n ] q, s has multiplicity g = [ n 2 ] q [ n ] q. Let a = v l (q + ), b = v l (qh ) and c = v l (h).

56 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2. r has multiplicity f = [ n ] q, s has multiplicity g = [ n 2 ] q [ n ] q. Let a = v l (q + ), b = v l (qh ) and c = v l (h). l h if and only if l n 2 2

57 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2. r has multiplicity f = [ n ] q, s has multiplicity g = [ n 2 ] q [ n ] q. Let a = v l (q + ), b = v l (qh ) and c = v l (h). l h if and only if l n 2 2 gcd(h, qh ) =, so either c = 0 or b = 0.

58 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2. r has multiplicity f = [ n ] q, s has multiplicity g = [ n 2 ] q [ n ] q. Let a = v l (q + ), b = v l (qh ) and c = v l (h). l h if and only if l n 2 2 gcd(h, qh ) =, so either c = 0 or b = 0. b = 0 if and only if l [ n 2 ] q.

59 Example: l q +, n even Assume l q +, n even. Then k = q(q + ) 2 h, r = (q + )(qh ), s = (q + ), where h := qn 2 q 2. r has multiplicity f = [ n ] q, s has multiplicity g = [ n 2 ] q [ n ] q. Let a = v l (q + ), b = v l (qh ) and c = v l (h). l h if and only if l n 2 2 gcd(h, qh ) =, so either c = 0 or b = 0. b = 0 if and only if l [ n 2 ] q. We ll look at the case c = 0 and b = 0. Then v l (r) = v l (s) = a and v l ( S(Γ) ) = af + ag + 2a

60 Example: l q +, n even, v l (r) = v l (s) = a M = Z l Y.

61 Example: l q +, n even, v l (r) = v l (s) = a M = Z l Y. On Y = Ker(J) we have A (A (r + s)i) = rsi.

62 Example: l q +, n even, v l (r) = v l (s) = a M = Z l Y. On Y = Ker(J) we have A (A (r + s)i) = rsi. This shows that (A (r + s)i)(y ) M 2a Y, so as l (r + s), A (Y ) M 2a Y.

63 Example: l q +, n even, v l (r) = v l (s) = a M = Z l Y. On Y = Ker(J) we have A (A (r + s)i) = rsi. This shows that (A (r + s)i)(y ) M 2a Y, so as l (r + s), A (Y ) M 2a Y. A (Y ) is nonzero, so it has D as a composition factor.

64 Example: l q +, n even, v l (r) = v l (s) = a M = Z l Y. On Y = Ker(J) we have A (A (r + s)i) = rsi. This shows that (A (r + s)i)(y ) M 2a Y, so as l (r + s), A (Y ) M 2a Y. A (Y ) is nonzero, so it has D as a composition factor. dim M 2a Y f and so dim M 2a f.

65 Example: l q +, n even, v l (r) = v l (s) = a M = Z l Y. On Y = Ker(J) we have A (A (r + s)i) = rsi. This shows that (A (r + s)i)(y ) M 2a Y, so as l (r + s), A (Y ) M 2a Y. A (Y ) is nonzero, so it has D as a composition factor. dim M 2a Y f and so dim M 2a f. Further analysis of A and structure of M shows dim M a = g + 2.

66 We have dim M 2a f, dim M a = g + 2.

67 We have dim M 2a f, dim M a = g + 2. a(f + g) + 2a = v l ( S(Γ) ) = ie i i 0 ie i + ie i a a i<2a a i<2a i 2a e i + 2a i 2a a(dim M a dim M 2a ) + 2a dim M 2a a(g + 2) + af. Therefore, equality holds throughout, and it follows that e 0 = f, e a = g f + 2, e 2a = f (and e i = 0 otherwise). e i

68 Critical groups of graphs Graphs from lines Smith normal forms Chip-firing game Cross-characteristics Defining characteristic

69 Here the number of composition factors of M grows like n t, where q = p t.

70 Here the number of composition factors of M grows like n t, where q = p t. But from the eigenvalues, L has no p-elementary divisors.

71 Here the number of composition factors of M grows like n t, where q = p t. But from the eigenvalues, L has no p-elementary divisors. For A, we can see that only k is divisible by p, so S(Γ ) is cyclic of order p t.

72 Here the number of composition factors of M grows like n t, where q = p t. But from the eigenvalues, L has no p-elementary divisors. For A, we can see that only k is divisible by p, so S(Γ ) is cyclic of order p t. The skew lines matrices A and L are much harder but much of the difficulty was handled in earlier work for the case n = 4 (Brouwer-Ducey-S.)

73 Here the number of composition factors of M grows like n t, where q = p t. But from the eigenvalues, L has no p-elementary divisors. For A, we can see that only k is divisible by p, so S(Γ ) is cyclic of order p t. The skew lines matrices A and L are much harder but much of the difficulty was handled in earlier work for the case n = 4 (Brouwer-Ducey-S.) Note A L mod (p 4t ), so just consider A.

74 Example Table : The elementary divisors of the incidence matrix of lines vs. lines in PG(3, 9), where two lines are incident when skew. Elem. Div Multiplicity

75 For n = 4 we have A(A + q(q )I) = q 3 I + q 3 (q )J,

76 For n = 4 we have A(A + q(q )I) = q 3 I + q 3 (q )J, For general n, we have A(A + q( [ ] n 2 q 2)I) = q 3[ n 3 ]q I + q3 [ n 3 ] q ( [ ) n ]q (q + 2) J, q +

77 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J)

78 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t.

79 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t. We have A Y [A Y qz I] = q 3 z 2 I, where z, z 2 are units in Z p.

80 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t. We have A Y [A Y qz I] = q 3 z 2 I, where z, z 2 are units in Z p. This equation implies:

81 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t. We have A Y [A Y qz I] = q 3 z 2 I, where z, z 2 are units in Z p. This equation implies:. A Y has no elementary divisors p i for i > 3t.

82 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t. We have A Y [A Y qz I] = q 3 z 2 I, where z, z 2 are units in Z p. This equation implies:. A Y has no elementary divisors p i for i > 3t. 2. e i = e 3t i 0 i < t.

83 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t. We have A Y [A Y qz I] = q 3 z 2 I, where z, z 2 are units in Z p. This equation implies:. A Y has no elementary divisors p i for i > 3t. 2. e i = e 3t i 0 i < t. 3. e i = 0 unless 0 i t or 2t i 3t.

84 Reductions using SRG equation p [ n 2 ] q so Z L 2 p = Z p Y, Y = Ker(J) A() = k = q 4[ ] n 2 2 q, so e 4t. We have A Y [A Y qz I] = q 3 z 2 I, where z, z 2 are units in Z p. This equation implies:. A Y has no elementary divisors p i for i > 3t. 2. e i = e 3t i 0 i < t. 3. e i = 0 unless 0 i t or 2t i 3t. 4. Once we have e i 0 i < t we have them all.

85 Reduction to point-line incidence A r,s := the skewness matrix of r-subspaces vs. s-subspaces of V. So A = A 2,2.

86 Reduction to point-line incidence A r,s := the skewness matrix of r-subspaces vs. s-subspaces of V. So A = A 2,2. A r,s A r, A,s (mod p t )

87 Reduction to point-line incidence A r,s := the skewness matrix of r-subspaces vs. s-subspaces of V. So A = A 2,2. A r,s A r, A,s (mod p t ) Hence e i (A r,s ) = e i (A r, A,s ) for i < t.

88 Reduction to point-line incidence A r,s := the skewness matrix of r-subspaces vs. s-subspaces of V. So A = A 2,2. A r,s A r, A,s (mod p t ) Hence e i (A r,s ) = e i (A r, A,s ) for i < t. The p-elementary divisors for A r, (and A,r ) were computed by Chandler-Sin-Xiang (2006).

89 Reduction to point-line incidence A r,s := the skewness matrix of r-subspaces vs. s-subspaces of V. So A = A 2,2. A r,s A r, A,s (mod p t ) Hence e i (A r,s ) = e i (A r, A,s ) for i < t. The p-elementary divisors for A r, (and A,r ) were computed by Chandler-Sin-Xiang (2006). Nontrivial to relate p-elementary divisors of A r, and A,s to those of A r, A,s.

90 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t,

91 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n,

92 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n, 0 ps i+ s i (p )n,

93 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n, 0 ps i+ s i (p )n, The poset H describes the submodule structure of F L q under the action of GL(n, q). (Bardoe-S (2000)).

94 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n, 0 ps i+ s i (p )n, The poset H describes the submodule structure of F L q under the action of GL(n, q). (Bardoe-S (2000)). For s = (s 0,..., s t ) H define d( s) as follows.

95 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n, 0 ps i+ s i (p )n, The poset H describes the submodule structure of F L q under the action of GL(n, q). (Bardoe-S (2000)). For s = (s 0,..., s t ) H define d( s) as follows. Set λ i = ps i+ s i (0 i t subscripts mod t).

96 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n, 0 ps i+ s i (p )n, The poset H describes the submodule structure of F L q under the action of GL(n, q). (Bardoe-S (2000)). For s = (s 0,..., s t ) H define d( s) as follows. Set λ i = ps i+ s i (0 i t subscripts mod t). d( s) = t i=0 d λ i, where d k := coefficient of x k in the expansion of ( + x + + x p ) n

97 The poset H H := the set of tuples { s = (s 0,..., s t ) such that, for 0 i t, s i n, 0 ps i+ s i (p )n, The poset H describes the submodule structure of F L q under the action of GL(n, q). (Bardoe-S (2000)). For s = (s 0,..., s t ) H define d( s) as follows. Set λ i = ps i+ s i (0 i t subscripts mod t). d( s) = t i=0 d λ i, where d k := coefficient of x k in the expansion of ( + x + + x p ) n d( s) is the dimension of a GL(n, q) composition factor of F L q.

98 For nonnegative integers α, β, define the subsets of H and { t } H α (s) = (s 0,..., s t ) H max{0, s s i } = α βh(r) = {(n s 0,..., n s t ) (s 0,..., s t ) H β (r)} { t } = (s 0,..., s t ) H max{0, s i (n r)} = β}. i=0 i=0

99 General formula for e i (A r, A,s ) Theorem Let E i = e i (A r, A,s ) denote the multiplicity of p i as a p-adic elementary divisor of A r, A,s. E t(r+s) =. For i t(r + s), where Γ(i) = E i = s Γ(i) α+β=i 0 α t(s ) 0 β t(r ) d( s), βh(r) H α (s).

100 Thank you for your attention!

Smith Normal Forms of Strongly Regular graphs

Smith Normal Forms of Strongly Regular graphs Peter Sin, U. of Florida UD Discrete Math. Seminar, May 7th, 2018. Smith normal forms Smith normal form Smith normal forms associated with graphs Smith and