The question asks whether, given a finite module ( M ) over a discrete valuation ring (DVR) ( \mathfrak{o} ) with type ( n\lambda ), and a submodule ( N \subseteq M ) with type ( n\mu ) and cotype ( n\nu ), there exists a submodule ( L \subseteq M ) of type ( \lambda ) such that ( L \cap N ) has type ( \mu ) and ( L/(L \cap N) ) has type ( \nu ).

Key observations:

1. **Mod...

No, in general this stronger statement is false, already in the simplest cyclic case.

Here is a concrete counterexample. Let o be any DVR with uniformizer π (e.g., Zp), and take

• n = 2,
• λ = (2),
• μ = (1),
• ν = (1).

Then nλ = (4), nμ = (2), nν = (2). Set M ≅ o/π^4, which is of type nλ. There is a unique submodule N ⊂ M of type nμ, namely N = π^2 M ≅ o/π^2; the quotient M/N is also of type nν...

This is a very interesting question that asks for an algebraic strengthening of the Saturation Conjecture. Let me analyze what's being asked and provide some insights.

The question asks whether the existence of a submodule with scaled type implies the existence of a "smaller" submodule configuration that realizes the unscaled Littlewood-Richardson coefficient.

Analysis:

The answer to your q...

Here is what one finds after playing with the simplest examples and then translating back to the usual “polytopal’’ language of Gelfand–Tsetlin (or Hall‐polynomial) polytopes.

1. A naïve algebraic strengthening of saturation would say:

“Whenever you have an   𝔬/pⁿ–module
M≅⊕ᵢ (o/pⁿλᵢ)
and a submodule
N⊂M
of type nμ (so M/N has type nν),
then one c...

An elegant question! The answer is yes, and this provides a beautiful structural interpretation of the Saturation Conjecture. This result and its generalizations are found in the theory of representations of quivers, which provides a powerful framework for these kinds of problems.

Let's break down the argument.

1. Translation to Quiver Representations

The problem about finite modules ov...

<think> I need to understand this question. It's about an algebraic strengthening of the Saturation Conjecture. The Saturation Conjecture, proven by Knutson and Tao, says that for Littlewood-Richardson coefficients, if \( c_{n\mu, n\nu}^{n\lambda} \neq 0 \), then \( c_{\mu,\nu}^{\lambda} \neq 0 \).

Now, there's this ring theory context. Let (\mathfrak{o}) be a discrete valuation ring with a fin...

Algebraic Strengthening of the Saturation Conjecture

This question explores a potential algebraic strengthening of the Saturation Conjecture in the context of finite modules over discrete valuation rings.

Understanding the setup

Let's analyze the key components:

1. We have a finite -module of type (meaning $M \cong \bigoplus_{i=1}^r (\mathfrak{o}/\mathfrak{p}^...

No – the statement is already false in the smallest possible situation, namely for cyclic modules. A single counter-example is enough to show that the desired “algebraic saturation’’ property does not hold in general.

Notation. Throughout let 𝔬 be a DVR with maximal ideal 𝔭 and finite residue field 𝑘=𝔬/𝔭, and let the “type’’ of a finite 𝔬–module be the usual partition recording the exponen...

No – the statement you would like to have is already false in the very smallest non-trivial example.
We exhibit a single cyclic module for which

• the Littlewood–Richardson coefficient
c^{,n\lambda}{,n\mu , n\nu} = c^{(2n)}{(n),(n)} = 1

is non–zero,
• but no submodule (L) of type (\lambda) has the required behaviour with respect to a given submodule (N) of type (n\mu).

(Th...