60,957 research outputs found
Some necessary conditions for vector space partitions
Some new necessary conditions for the existence of vector space partitions
are derived. They are applied to the problem of finding the maximum number of
spaces of dimension t in a vector space partition of V(2t,q) that contains m_d
spaces of dimension d, where t/2<d<t, and also spaces of other dimensions. It
is also discussed how this problem is related to maximal partial t-spreads in
V(2t,q). We also give a lower bound for the number of spaces in a vector space
partition and verify that this bound is tight.Comment: 19 pages; corrected typos and rewritten introductio
Interference alignment for the MIMO interference channel
We study vector space interference alignment for the MIMO interference
channel with no time or frequency diversity, and no symbol extensions. We prove
both necessary and sufficient conditions for alignment. In particular, we
characterize the feasibility of alignment for the symmetric three-user channel
where all users transmit along d dimensions, all transmitters have M antennas
and all receivers have N antennas, as well as feasibility of alignment for the
fully symmetric (M=N) channel with an arbitrary number of users.
An implication of our results is that the total degrees of freedom available
in a K-user interference channel, using only spatial diversity from the
multiple antennas, is at most 2. This is in sharp contrast to the K/2 degrees
of freedom shown to be possible by Cadambe and Jafar with arbitrarily large
time or frequency diversity.
Moving beyond the question of feasibility, we additionally discuss
computation of the number of solutions using Schubert calculus in cases where
there are a finite number of solutions.Comment: 16 pages, 7 figures, final submitted versio
Generalized Littlewood-Richardson coefficients for branching rules of GL(n) and extremal weight crystals
Following the methods used by Derksen-Weyman in \cite{DW11} and Chindris in
\cite{Chi08}, we use quiver theory to represent the generalized
Littlewood-Richardson coefficients for the branching rule for the diagonal
embedding of \gl(n) as the dimension of a weight space of semi-invariants.
Using this, we prove their saturation and investigate when they are nonzero. We
also show that for certain partitions the associated stretched polynomials
satisfy the same conjectures as single Littlewood-Richardson coefficients. We
then provide a polytopal description of this multiplicity and show that its
positivity may be computed in strongly polynomial time. Finally, we remark that
similar results hold for certain other generalized Littlewood-Richardson
coefficients.Comment: 28 pages, comments welcom
Sequences defined by h-vectors
In this paper we consider the sequence whose n^{th} term is the number of h-vectors of length n. We show that the n^{th} term of this sequence is bounded above by the n^{th} Fibonacci number and bounded below by the number if integer partitions of n into distinct parts. Further we show embedded sequences that directly relate to integer partitions
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