Group Actions on Posets and Cartesian Products Anton Betten Colorado State University June 2014
Title: Group Actions on Posets and Cartesian Products Abstract: Classifying combinatorial objects like graphs and other things requires us to compute orbits of groups acting on partially ordered sets.
Our Goals We wish to classify combinatorial objects. Never mind that some of these structures have been called “unclassifyable” (Jack Koolen)
Our Goals • Graphs • Tournaments • Designs • Codes • Structures from Geometry (Spreads, Packings, BLT-Sets, Hyperovals etc.) • Structures from Algebra (Semifields) (fill in your favorite structure • here)
Our Goals Classification means that we find all the pairwise non-isomorphic objects. Isomorphic objects are objects that lie in the same orbit of the symmetry group of the space. Recognition means that we find can identify the orbit containing an arbitrary object (in a previously constructed classification). Constructive Recognition means that we can recognize the orbit containing a given element x and we can find a group element g 2 G that maps x to the orbit representative.
Q: What kind of answer do we expect? It depends. Note that the output is exponential, so we cannot have a polynomial algorithm (in the size of the input). However, we strive to provide efficient algorithms . An efficient algorithm is one that solves small instances of the problem at hand in reasonable time .
In a partially ordered set ( P , � ) , x � y means x is contained in y (or x is a subobject of y ) A group G acts on P if x g � y g ( ) x � y A partially ordered set P is a lattice if any two elements x , y in P have a supremum x _ y and an infimum x ^ y .
Let’s see an example: Example 1: Suppose we want to classify the graphs on n vertices. Take a set V of size n . Consider the set E of all unordered pairs from V (edges). A graph on V is a subset of E . The group Sym V acts on V . It induces an action on E and hence also in graphs on V .
The poset: The set of graphs on V is ordered by inclusion: (i.e., subobjects are subgraphs). This poset is a lattice: x _ y is the “union graph” and x ^ y is the intersection graph.
Here are some small cases: (note that we draw the lattice “upside down,” this is the way people in Computer Science like it):
The Graphs on 3 Vertices 8 labeled graphs that fall into 4 isomorphism classes.
The Graphs on 4 Vertices 64 labeled graphs that fall into 11 isomorphism classes.
What we really want is not the lattice but the poset of orbits : Nodes correspond to isomorphism types of graphs (orbits). Edges are between two orbits for which at least one pair of elements is related.
The Graphs on 4 Vertices
We want to classify a class of graphs defined by some extra property. In these cases, we require that the poset of those graphs is a semilattice (i.e., closed under intersections).
Example 2 (Regular graphs): Let’s classify graphs that are regular of degree d . The poset is the set of subsets of E with the property: Each vertex v 2 V is incident with at most d elements in the chosen set of edges.
The 2-Regular Graphs on 4 Vertices
The 2-Regular Graphs on 6 Vertices
The Search Tree The search tree results from the poset of orbits by removing edges: Let’s remove all but one incoming edge per node. Let’s always keep the edge from the leftmost predecessor. This is the idea of the “canonical predecessor” (McKay 1998).
The 2-Regular Graphs on 6 Vertices
The 3-Regular Graphs on 6 Vertices Interestingly, the 3-Regular Graphs on 6 Vertices are much more difficult to compute. (even though we know that there are exactly two, the complements of the ones we had before).
The 3-Regular Graphs on 6 Vertices
The Classification Algorithm Examine the search tree arising from the semilattice P . Proceed from bottom to top and compute the orbits of the symmetry group G on P . Q: How?
Some possibilities: • Use of canonical forms (lex-least element in an orbit). • Use of canonical predecessors (to make sure each orbit is listed exactly once). • Use the stabilizer subgroup of partial objects to reduce the search space below. Many of these ideas are based on work of • Read • Faradjev • McKay There is the questions of depth-first search vs. breadth-first search.
Another idea is due to Schmalz 1992 (Ph.D. with Laue): Store some information in a data structure. (Retrieval may be faster than recomputing all the time). To explain what we actually store, we need to look at some theory.
Let X and Y be two finite sets. Let the group G act on both X and on Y . Let I be a G -invariant relation between X and Y : ) ( x g , y g ) 2 I . ( x , y ) 2 I (
For a 2 X , let Up ( a ) = { ( x , y ) 2 I | x = a } . For b 2 Y , let Down ( b ) = { ( x , y ) 2 I | y = b } . Suppose that P 1 , . . . , P m is a transversal for the G -orbits on X . Suppose that Q 1 , . . . , Q n is a transversal for the G -orbits on Y .
The Stabilizer G a of a 2 X acts on Up ( a ) . For i = 1 , . . . , m , let U i , 1 , . . . , U i , m i be a transversal for the orbits of G P i acting on Up ( P i ) . Let U = { U i , s | i = 1 , . . . , m , s = 1 , . . . , m i }
The Stabilizer G b of b 2 Y acts on Down ( b ) . For j = 1 , . . . , n , let D j , 1 , . . . , D j , n j be a transversal for the orbits of G Q j acting on Down ( Q j ) . Let D = { D j , t | j = 1 , . . . , n , t = 1 , . . . , n j }
LEMMA: There is a canonical bijection between the elements of the set U and the elements of the set D . In particular, m n X X m i = n j . i = 1 j = 1 Proof: Both sets are in one-to-one correspondence to the orbits of G on the elements of I . Notation: Let Φ be the canonical bijection arising from the Lemma.
Underpinning the mapping Φ is another mapping ϕ : U ! G such that for all U 2 U and g = U ϕ : U Φ = D U g 7! D . )
Classify group orbits on (ranked) posets by induction. Let X be the objects at level i . Let Y be the objects at level i + 1. Let I be the relation between X and Y given by the poset structure: ( x , y ) 2 I ( ) x � y
Assume P 1 , . . . , P m are known (orbit representatives on X ). Assume G P i are known (the associated stabilizer subgroups). For i = 1 , . . . , m , compute a transversal U i , 1 , . . . , U i , m i for the orbits of G P i on Up ( P i ) .
Let C = { Π 2 ( U ) | U 2 U} (here Π 2 is the projection onto the second component). Pare down the list C to the desired list Q 1 , . . . , Q n of representatives on Y . At the same time, compute the stabilizers G G j .
To do so, use the set Down ( c ) for c 2 C and use the induction hyphothesis (constructive recognition) to remove any element in C that is isomorphic to c . Let Φ : U ! D be the canonical bijection, let ϕ : U ! G be the mapping described above. Example: The classification of graphs on 4 vertices (revisited):
The Graphs on 4 Vertices
The Graphs on 4 Vertices Let’s take a closer look: We focus on level 2 versus level 3: 1 2 3 4 The edges in the search poset are labeled by pairs of orbits from U and from D that correspond under the bijection φ .
The Graphs on 4 Vertices: Levels 2 and 3 P 1 P 2 U 1 , 1 U 1 , 2 U 1 , 3 U 2 , 1 ϕ = id ϕ = id ϕ = id ϕ = ( 2 , 3 ) D 1 , 1 D 2 , 1 D 3 , 1 D 3 , 2 Q 1 Q 2 Q 3
The Graphs on 4 Vertices: Levels 2 and 3 So, under the canonical bijection Φ : U 1 , 1 $ D 1 , 1 U 1 , 2 $ D 2 , 1 U 1 , 3 $ D 3 , 1 U 2 , 1 $ D 3 , 2 So, we have additional “middle levels” corresponding to the flag orbits of G :
The Graphs on 4 Vertices: Auxillary Poset
Summary The data that we store is essentially the maping ϕ . It is in this regard that Schmalz’s algorithm differs from McKay’s and Read’s and Faradjev’s.
Finite Semifields Roughly speaking: a semifield is a field without associativity. Semifields are classified up to isotopy. Here is the semifield of order 8: 1 0 0 0 0 1 0 1 0 0 1 0 1 0 1 0 1 1 , , 0 0 1 0 1 0 1 0 1 Any nonzero linear combination of the basis matrices is invertible.
Classification Semifields of order 16 were classified in 1960 by Bruck and Kleinfeld. Semifields of order 32 were classified in 1965 by Knuth. Consider the poset of subspaces of matrices with the property that any nonzero linear combination is invertible. This is a semilattice.
Semifields of Order 16 The poset of G -orbits on the semilattice is: 20160 30 72 1080 6 18 7 4 24 60 168 18 108 900
Order 64 was done in 2009 by Rua, Combarro and Ranilla. However, there have been some questions about the correctness of the result. So, we are currently running an independent check. We expect that this will require around 25 years CPU-time. We will be using the Open Science Grid. Thank You for Your Patience!
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