The Meaning of a Permutation

The Meaning of a Permutation PhL 11.16.15 What does one mean by a permutation? Suppose P{1,2,3...k} = {i1, i2, ....ik} = {P(1),P(2),...P(k)} P(r) = ir Notice that we can talk about P(z) where P acts on a set, or P(3) where P acts on a set element. P(z0) = iz0 // a possible notation for the above. Now consider our example, P[ T12...k] = TP(1)P(2)...P(k) = Tii...i No ambiguity to this point. Now suppose Q{1,2,3,4} = {j1, j2, j3, j4 } = {Q(1),Q(2),Q(3),Q(4)} Q(r) = jr What could you say about Q{1,3,2,4} ? Q{1,3,2,4} = {Q(1),Q(3),Q(2),Q(4) } = {j1, j3, j2, j4 } Still there seems to be no ambiguity. What could you say about Q{i1, i2, i3, i4} ? Q{i1, i2, i3, i4} = {Q(i1),Q(i2),Q(i3),Q(i4)} For example, Q{1,2,3,4} = {Q(1),Q(2),Q(3),Q(4)} = {j1, j2, j3, j4 } Big Question #1. Is it correct to say that Q(ir) = iQ(r) ?? First of all, if this WERE true to say, then we would have, Q{i1, i2, i3, i4} = {Q(i1),Q(i2),Q(i3),Q(i4)} = {iQ(1),iQ(2),iQ(3),iQ(4)} = {ij1, ij2, ij3, ij4} Let's test this idea with a specific case and see what happens. Suppose Q{1,2,3,4} = {Q(1),Q(2),Q(3),Q(4)} = {1,4,3,2} = {j1, j3, j2, j4 } We would naturally say, using only the above line, Q{1,3,2,4} = {Q(1),Q(3),Q(2),Q(4)} = {1,3,4,2} Now try our hypothetical formula, Q{i1, i2, i3, i4} = {iQ(1),iQ(2),iQ(3),iQ(4)} Apply this with {i1, i2, i3, i4} = {1,3,2,4} Q{1,3,2,4} = {iQ(1),iQ(2),iQ(3),iQ(4)} = {i1, i4, i3, i2} = {1,4,2,3} This disagrees with the "naturally obtained" result above which said this: Q{1,3,2,4} = {1,3,4,2} Why are these different? In the first case, Q permutes the actual items in the list. In the second case, Q permutes the subscripts on the items in the list. These two notions of permutation are completely different!!! If we decide to use the second meaning, then P{1,2,3,4} = undefined, because there are no subscripts to permute! How can we avoid confusing these two notions? How would you "set up shop" to use the second meaning? Here is an idea, P{1,2,3..k} = {P(1),P(2), ....P(k)} = as usual P{i1, i2, i3, i4} = {P(i1),P(i2),P(i3),P(i4) } // permuting list elements P[i1, i2, i3, i4] ≡ {iP(1), iP(2),iP(3), iP(4)} // permuting subscripts of list elements This new P[...] thing means you are permuting subscripts. Now go back to this idea, f[z] = f[1,2....k] = Tii...i = T(vi,vi, .... vi) I THINK I want to permute subscripts of labels, not the labels themselves/ For example, the labels might not be in the range 1 to k. So, with this subscript meaning of permutation, consider P f[1,2....k] = f[P(1),P(2).....P(k)] = Tii...i Suppose our meaning is that a permutation acts on the numbers 1,2,3..k wherever they are. Then the above would be correct. But then what is this? Q P f[1,2....k] = Q f[P(1),P(2).....P(k)] We make Q act directly on the numbers 1,2,3...k so we get Q P f[1,2....k] = Q f[P(1),P(2).....P(k)] = P(Q(1)), P(Q(2)).... P(Q(k)) And then Q P f[1,2....k] = Tii...i Then R Q P f[1,2....k] = R Q f[P(1),P(2).....P(k)] = R f[P(Q(1)),P(Q(2)).....P(Q(k))] = f [PQR(1), PQR(2), ....PQR(k) ) This is what I would LIKE to happen, but how can I justify it? Put this more in context as in Lemma of Appendix C original. f[1,2....k] = ΣQ(-1)S(Q) F[Q(1),Q(2)....Q(k)] Now suppose I try this (without justification right now) P F[Q(1),Q(2)....Q(k)] = F( P(Q(1)), P(Q(2)).... P(Q(k)) ) = F( PQ(1), PQ(2).... PQ(k) ) Am I saying that P "acts on the positions" ? Then we get P f[1,2....k] = ΣQ(-1)S(Q) P F[Q(1),Q(2)....Q(k)] = ΣQ(-1)S(Q) F( PQ(1), PQ(2).... PQ(k) ) with summation index on the right where I NEED it to be. = Q Tii...i What does Q act on? We said earlier that it acts on the subscripts, and here subscripts are like P(1) so then, Q P f[1,2....k] = Q f[P(1),P(2).....P(k)] = Q Tii...i = Tii... = Tii...i But this is not the order than I need!