kevin-idea.rtf

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Hi, Yan. \'a0I haven\'92t forgotten your sketch of an article. \'a0I\'92ve been thinking about a new way of organizing my thoughts on 2-adinkras, and I thought I would run it by you. \'a0I think this might simplify our proofs.\
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Given a connected 2-Adinrka, A, with m left-moving colors and n right-moving colors, we can form the 1-Adinkra A_L with just the m left-moving colors and A_R with just the n right-moving colors (I know this conflicts with other notation we have used). \'a0\'a0Both A_L and A_R are unions of connected components, of course.\
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Proposition: If X is a connected component of A_L, then the bigrading of the vertices (x,y) has constant value for y.\
[Likewise, connected components of A_R have constant value for x]\
Proof: given any two vertices, there is a path with only left-moving colors, and this keeps y constant.\
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Definition: for each right-moving color, i=1,\'85, n, define q_i: vertex(A_L) -> vertex(A_L), to be the result of following right-moving color i, where vertex(A_L) is the vertex set of A_L.\
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Proposition: if X is a connected component of A_L, and i is any right-moving color, then q_i(X) is the vertex set of a connected component of A_L.\
[Likewise for using left-moving colors on connected components of A_R]\
Proof: given two vertices q_i(a) and q_i(b) in q_i(X), we have that a and b are in X. \'a0By connectedness of X there is a path using left-moving colors from a to b, and so a sequence of Q_I connects them. \'a0By anticommutativity with Q_i, we get that this connects q_i(a) to q_i(b).\
Conversely, suppose q_i(a) is connected with left-moving colors to c. \'a0Then applying Q_i we get a path of left-moving colors from a to q_i(c) which we call b. \'a0Then c=q_i(b) where b is in X.\
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Proposition: If X is a connected component of A_L, with y equal to the constant y value in the bigrading (x,y), and if i is a right-moving color, then q_i(X) either has constant y-value y+1 or has constant y-value y-1.\
Proof: \'a0q_i connects vertices of bigrading (x,y) to vertices of bigrading (x,y+1) or (x,y-1). Since q_i(X) is a connected component of A_L, then it has constant value for the y-value. \'a0Therefore all the vertices in q_i(X) either all have y-value y+1 or all have y-value y-1.\
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Theorem: If X and Y are connected components of A_L, then there is an integer k and a map q:X -> Y that is an isomorphism of graphs that adds (0,k) to every bigrading of every vertex of X.\
Proof: Pick a point in X and a point in Y. \'a0Since the original 2-Adinkra A was connected, there is a path connecting the points so chosen. \'a0Re-order the path to be all right-moving colors then all left-moving colors. \'a0We then have a sequence of right-moving colors forming a path from a point in X to a point in Y. \'a0Define q to be the composition of the maps q_i for each of the right-moving colors in the sequence. \'a0The previous proposition then gives us the result.}