Mathematics 676
Homework (due Apr 28)
A. Hulpke
52) What is “wrong” with the following pc-presentation? (What order do you “expect” from the presentation?)
⟨a, b ∣ a 2 = b5 = 1, b a = b2 ⟩
53) Let M be the monoid with the presentation ⟨x, y ∣ x 3 = 1, y3 = 1, (x y)3 = 1⟩.
a) Determine (by hand) a confluent rewriting system for M with respect to length+lexicographic ordering.
b) Using the rules determined in a), construct an infinite set of words in normal form, thus proving that M
is infinite.
54) A Dynkin diagram is a graph on the vertices {1, . . . , n}, whose edges {i, j} are labelled with weights
e i, j ∈ {3, 4, . . .} (often, weight k is indicated by drawing the edge in multiplicity k − 2, i.e. no edge would be
weight 2, which is consistent with the rest of the definition). We set e i, j = 2 if i and j are not connected.
We now associate to such a diagram on the vertices 1, . . . , n a finitely presented group (called the “Weyl group”
or “Coxeter group” for this diagram) on n generators a1 , . . . , a n with the relations: a 2i = 1 and for i < j and
a j ai a j ai ⋯ = ai a j ai a j⋯
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
e i, j letters e i, j letters
a) We now consider the Dynkin-diagram called “A n ”, which has the form 0 − 0 − 0 − ⋯ − 0 with n vertices.
So for example for n = 3 we get the presentation
⟨a1 , a2 , a3 ∣ a12 = a22 = a33 = 1, a2 a1 a2 = a1 a2 a1 , a3 a1 = a1 a3 , a3 a2 a3 = a2 a3 a2 ⟩
Show that the Weyl group defined by this diagram is isomorphic to S n+1 .
b) Show that the rules for the Dynkin diagram of type A n yield a confluent rewriting system for len/lex
ordering.
c) Give an example of a Dynkin diagram, whose presentation does not define a confluent rewriting system.
55) If you define a finitely presented group in GAP, you can enforce reduction of elements to normal form
(running a length/lex Knuth-Bendix algorithm on the presentation to obtain a confluent rewriting system)
by calling SetReducedMultiplication(G); Why do you think this is not turned on by default?
56) a) Why is a pure lexicographic ordering not a reduction ordering? Give a (counter)example.
b) Prove that the wreath product ordering of reduction orderings is again a reduction ordering.
57) Let G = ⟨a, b, c, d ∣ a 3 = 1, b2 = d, c 2 = d, d 2 = 1, b a = c, c a = bc, c b = cd, d a = d, d b = d, d c = d⟩. Use these
relations (which form a pc-presentation) to write the word dbcdab in normal form (i.e. letters in alphabetical
order and exponents bounded by the relative orders).