Matrix pairs over valuation rings and ℝ-valued Littlewood–Richardson fillings

Abstract

for any index set I of length k. For this, we will expand LD b ν Π r T L T U according to the Cauchy-Binet formula:Clearly this minimum is obtained when LD b ν IU is at a minimum, and the other factors have order zero. This can be accomplished, using Corollary 3.12, with the term

Hives Determined by Pairs in the Affine Grassmannian over Discrete Valuation Rings

“…In our earlier work, we were able to give, from a matrix pair (M, N) over a certain valuation ring, a hive construction of both types (µ, ν, λ) and of (ν, µ, λ). Further, we were able to show [2] (by means of a rather delicate argument) that the bijection c λ µν ↔ c λ νµ we constructed matricially exactly matched the combinatorially defined bijection (as described by James and Kerber [6]) known previously. Our Theorem 1.1 here seems likely to construct such a bijection between hives of type (µ, ν, λ) and (ν, µ, λ) and, indeed, to agree with our previous construction in [2], at least over the rings for which that earlier construction applied.…”

“…Further, we were able to show [2] (by means of a rather delicate argument) that the bijection c λ µν ↔ c λ νµ we constructed matricially exactly matched the combinatorially defined bijection (as described by James and Kerber [6]) known previously. Our Theorem 1.1 here seems likely to construct such a bijection between hives of type (µ, ν, λ) and (ν, µ, λ) and, indeed, to agree with our previous construction in [2], at least over the rings for which that earlier construction applied. The function would map a hive {h st } of type (µ, ν, λ) to a hive {h ′ st } of type (ν, µ, λ) provided there is a pair of (N , Λ) ∈ Gr × Gr of the appropriate type for which both hive constructions of Theorem 1.1 applied to (N , Λ) yield the hives {h st } and {h ′ st }.…”

“…The first author's work [1] constructed matrix realizations (M, N) from an arbitrary Littlewood-Richardson filling of type (µ, ν, λ). In the authors' work [2] these results were extended to matrix pairs (M, N ) in GL n (K) by defining Littlewood-Richardson fillings admitting some negative, real-valued entries, also showing that this matrix setting realized combinatorial bijections establishing c λ µν = c λ νµ . In Kamnitzer [7] these issues were formulated in terms of elements of the affine Grassmannian Gr over O = C[[t]], where from each pair (N , Λ) ∈ Gr a hive (defined below) was determined, among other results.…”

“…However, the sixth condition in the right-leaning rhombus inequality fails. Namely, the blocks Λ A n−j+1 (2) and Λ C n−j+1 (2) do not lie along the same row (and, in this case, both are of the same rank). However, what we do see is that in the proposed matricial inequality:…”

“…, namely α 1 , cannot exceed the largest invariant factor of Λ C n−j+1 (2) since it must actually be among the invariant factors of Λ C n−j+1 (2) . With this, the last inequality has been verified, and the proof is complete.…”