8 1. THE CALCULUS OF SWEEDLER POWERS (cf. [17], Par. 8.5, p. 474 [50], Sec. 1, p. 3). But the introduction of the additional permutation causes Sweedler powers to be closed under iteration, as the following power law for Sweedler powers asserts: PROPOSITION. For a e S n , r e 5 m , and h e H, we have (ha)T = har. PROOF. We give a simplification of the original argument that was pointed out by P. Schauenburg. Recall that a r satisfies by definition the equation a r((i - l)n + j) = (a(j) - l)m + r(i) We have in general that n n m (g) Am(hU)) = h{l) 0 ... 0 ft(mn) = (g) (g) /i((i_1)m+i) . = 1 j = l i=l We therefore get n n n m Am{hn = A m ( r j V w ) ) ) = -Q A ( V w ) ) = rj 0 /i ((T0 -)_ 1)m+) m n =0ri f t (wj)- i ) m + i ) i = l j = l Permuting the tensor factors and multiplying, we get ran m n mn (ha)T = J J J | h ( ( 7 y ) _ 1 ) m + r ( i ) ) = J J J l ft(cr.r((i-l)n+j)) = 1 1 ft (r.r(i)) = ^ 2=lj' = l i = l j = l 2=1 as asserted. D 1.4. We proceed to construct certain Sweedler powers from special permutations that are based on finite sequences of positive integers. We denote such sequences, using square brackets, in the form [rii,n 2 ,... ,rifc]. We call such a sequence nor- malized if every entry divides its predecessor, so that we have nk/rik-i/ . /ft2A&i. Given a sequence [ni, n 2 ,... , n^], we define its normalization [n^, n 2 ,... , n^] recur- sively as follows: We set n'x := ni and nj + 1 := gcd(ni+i,n£), the greatest common divisor of n2+i and nj. We define a product of two such sequences by the formula [rai, n 2 ,... , njfc][mi, m 2 ,... , mi] = [mini, min 2 ,... , minfc, mi, ra2,..., mi] In addition, we introduce a unique element, called the empty sequence and denoted by [ ], that is by definition normalized and a unit for this product. If we denote the set of all such sequences by M and the set of all normalized sequences by M^, we have the following result: PROPOSITION. M is a monoid and M^ is a submonoid. Normalization defines a monoid homomorphism from M to M^. PROOF. The product is associative since we have ([ni,n 2 ,... ,Wfc][mi,m2,... ,mi])[pi,p2,... ,pq] = \piminupim1n2, •. ,Pim1nk,p1mi,pim2,... ,Pim/,pi,p 2 ,... ,pq] = [ni, rc2,..., rife]([mi, m 2 ,... , mj][pi,p2, ,pq])
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