# A 2-groupoid Characterisation of the Cubical Homotopy - download pdf or read online By Harde K. A.

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E. g. 2 ~X [3,2,1 ]=; X /~ yields X X X X [(3,2,1 2 )']= X X =[4,2,1]. X Partitions a and Young diagrams [a] where a = a' are called self-associated. 4 P(n): = {yly~n} of all the partitIOns of n. ;;;; is a total order, so that the order diagram is always linear. 6 In the case when this holds we say that f3 dominates a and call ~ the dominance order. ) It is easy to see that the dominance order differs from the lexicographic order on pen) if and only if n~6. 8 a~f3 = a';;;;f3. J{3: [a

9 is already the complete set of these equivalence classes. 10 LEMMA. The multiplicity i(IS" iSn ,[,8]) of [,8] in IS" iSn is nonzero only if a~,8. 104 shows that i(ASw i Sn' [,8]) = 1. The assumption that i(IS" i SII' [,8])*0 yields therefore i(IS" i SII' ASf3 , 1SII )*0. Hence there exist 0-1 matrices with row sums a I and column sums p} f)I, which implies a ~,8. • Notice that in this proof we did only use the trivial part of the Gale-Ryser theorem, as we promised at the end of Section 104. We are now in a position to prove the main theorem of this section.

K( 1- Xi Yk ) - I. Since 1S(I") i Sn = RSn (d. 14 M= n * o Furthermore the scalar product of the first row of M n with its ith row is just the index ISn: Sa'l, l";;;;i";;;;p(n). But the knowledge of M n alone does not suffice to evaluate Zn' We also need :::n' the matrix of the permutation characters. Since 1Sa i Sn is the permutation representation of Sn on the left cosets of Sa it is in principle possible to evaluate ~a( 17) by checking which left cosets of Sa remain fixed under left multiplication by 17.