Invariants of matrix pairs over discrete valuation rings and Littlewood–Richardson fillings

Flows on honeycombs and sums of Littlewood–Richardson tableaux

“…It has been shown [6] that the same triples of partitions (possibly real-valued) appear in this matrix setting (possibly using a ring with real valuation) as those appearing in the Hermitian case. In the valuation ring setting the present authors have determined methods to associate a Littlewood-Richardson filling to matrix pairs, and conversely [1,2,3]. Furthermore, calculations by the present authors corroborate that Littlewood-Richardson fillings associated to the direct sum of matrices in the valuation ring context do, in fact, correspond to the sum of the fillings of the summands, as calculated by our algorithm here.…”

“…Word violations correspond to Type (1) non-canonical flows (see Figure 31). Column-strict violations correspond to Type (2). As such, these errors are corrected in exactly the same manner as a µ-flow (another Type (1) error) and "i in row i" (a Type (2) error).…”

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

“…Our central matrix result is the following determinantal formula that calculates a LR filling from a matrix pair. Most of the proof of this result follows exactly the argument presented in [2] in the case of a discrete valuation ring. We will sketch the proof and mention the few cases where passage to the R-valuation introduces changes.…”

“…In [2] the theory of µ-generic matrices was presented in the context of discrete valuation rings. While most of what was developed there works for R-valuation rings with few, if any, changes, we will need more general results in later sections.…”

“…The proof of the corollary below may be found in [2]. There, the results are stated for the case that R denotes a discrete valuation ring whose residue field is characteristic zero.…”