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Lectures
Seminar information archive ～06/25｜Next seminar｜Future seminars 06/26～
2012/07/23
16:30-17:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Thomas W. Roby (University of Connecticut)
Combinatorial Ergodicity (ENGLISH)
Thomas W. Roby (University of Connecticut)
Combinatorial Ergodicity (ENGLISH)
[ Abstract ]
Many cyclic actions tau on a finite set S of
combinatorial objects, along with many natural
statistics phi on S, exhibit``combinatorial ergodicity'':
the average of phi over each tau-orbit in S is
the same as the average of phi over the whole set S.
One example is the case where S is the set of
length n binary strings a1dotsan
with exactly k 1's,
tau is the map that cyclically rotates them,
and phi is the number of \\textit{inversions}
(i.e, pairs (ai,aj)=(1,0) for iJ less than j).
This phenomenon was first noticed by Panyushev
in 2007 in the context of antichains in root posets;
Armstrong, Stump, and Thomas proved his
conjecture in 2011.
We describe a theoretical framework for results of this kind,
and discuss old and new results for products of two chains.
This is joint work with Jim Propp.
Many cyclic actions tau on a finite set S of
combinatorial objects, along with many natural
statistics phi on S, exhibit``combinatorial ergodicity'':
the average of phi over each tau-orbit in S is
the same as the average of phi over the whole set S.
One example is the case where S is the set of
length n binary strings a1dotsan
with exactly k 1's,
tau is the map that cyclically rotates them,
and phi is the number of \\textit{inversions}
(i.e, pairs (ai,aj)=(1,0) for iJ less than j).
This phenomenon was first noticed by Panyushev
in 2007 in the context of antichains in root posets;
Armstrong, Stump, and Thomas proved his
conjecture in 2011.
We describe a theoretical framework for results of this kind,
and discuss old and new results for products of two chains.
This is joint work with Jim Propp.
