Fanions: Flagged Schur & Matrix Updates

• Fanions are a dual-use concept: in algebraic combinatorics they denote flagged Schur functions, and in optimization they refer to matrix update methods using Ky Fan norms.
• In combinatorics, flagged Schur functions are defined through tableau constraints and are enumerated by parabolic Catalan numbers, linking them to Demazure characters and convex polytopes.
• In optimization, Fanion algorithms interpolate between low-rank and full-rank updates, providing robust performance improvements in tasks such as LLM training and linear regression.

Fanions are a concept appearing in several specialized mathematical contexts. In algebraic combinatorics, "fanion" denotes flagged Schur functions, which provide a refined class of symmetric polynomials with explicit combinatorial and representation-theoretic significance. In modern optimization for large-scale learning, "Fanions" refer to a parameterized family of matrix update algorithms that interpolate between low-rank and full-rank steps using Ky Fan norms and their convex combinations. Both usages are rooted in deep mathematical structures and have concrete theoretical and algorithmic impact.

1. Flagged Schur Functions: Definition and Tableau Structure

Let $n\ge1$ be an integer and let $$\lambda=(\lambda_1\ge\lambda_2\ge\dots\ge\lambda_n\ge0)$$ be a partition with at most $n$ parts. For such $\lambda$, a "flag" is an $n$-tuple $\varphi=(\varphi_1,\dots,\varphi_n)$ satisfying $1\le\varphi_1\le \dots \le\varphi_n\le n$. The set $\mathcal T_\lambda$ comprises semistandard Young tableaux of shape $\lambda$ with entries in $[n]$—weakly increasing across rows, strictly increasing down columns.

For $T\in\mathcal T_\lambda$, let $\Theta(T)=(\theta_1,\dots,\theta_n)$ with $\theta_i$ the multiplicity of entry $i$ in $T$, so $$x^{\Theta(T)} = x_1^{\theta_1}\cdots x_n^{\theta_n}$$. The flagged Schur function indexed by $(\lambda,\varphi)$ is

$$s_\lambda(\varphi;x) = \sum_{T \in \mathcal S_\lambda(\varphi)} x^{\Theta(T)},\quad \mathcal S_\lambda(\varphi)=\{T\in\mathcal T_\lambda : T_{i,j}\le\varphi_i\},$$

that is, the sum over $T$ such that every entry in row $i$ lies in $[i,\varphi_i]$ (Proctor et al., 2016). The tableau constraint enables highly structured combinatorics, with applications to Demazure characters and polynomial representations.

2. Enumeration and Parabolic Catalan Numbers

Enumeration of flagged Schur functions is governed by parabolic Catalan numbers. For $R\subseteq\{1,\dots,n-1\}$, define $R=\{q_1<\dots<q_r\}$, with $q_0=0$, $q_{r+1}=n$. An $R$-permutation is a permutation $\pi\in S_n$ increasing on each block ("carrel") $(q_{h-1},q_h]$. Such a permutation is $R$–312–avoiding if it contains no pattern of the type $\pi_a>\pi_b<\pi_c$ with $a\le q_h< b\le q_{h+1}<c\le n$ and $\pi_a>\pi_c$.

The parabolic $R$–Catalan number

$$C_n^R = |\{\pi\in S_n^R: \pi \text{ is } R\text{–312–avoiding}\}|$$

counts not only $R$–312–avoiding $R$–permutations, but also enumerates: the set of distinct flagged Schur functions $s_\lambda(\varphi;x)$ with $R$ as the shape's set of non-maximal column lengths, gapless $R$–tuples, $R$–floor and $R$–ceiling flags, $R$–rightmost clump–deleting chains, and combinatorial keys (Proctor et al., 2016).

3. Convexity of Demazure Sets and Coincidences

For $\pi\in S_n^R$, the Demazure character (key polynomial) is defined as

$$d_\lambda(\pi;x)=\sum_{T\in\mathcal D_\lambda(\pi)} x^{\Theta(T)},$$

where $\mathcal D_\lambda(\pi)$ is the set of $\pi$–Demazure tableaux. These sets in $\mathbb Z^{|\lambda|}$ are convex polytopes (principal ideals $[Y_\lambda(\pi)]$) if and only if $\pi$ is $R$–312–avoiding. The scanning-tableau method realizes $\mathcal D_\lambda(\pi) = \{S\le Y_\lambda(\pi)\}$ (Proctor et al., 2016).

Flagged Schur functions and Demazure characters coincide precisely when their indexing sets correspond under canonical maps. Specifically, $s_\lambda(\varphi;x)=d_\lambda(\pi;x)$ if and only if $\pi$ is $R$–312–avoiding and the indexing flags correspond via established bijections. This coincidence is unique at the level of tableau sets.

4. Fanions in Matrix Optimization: Ky Fan Norms and Dual Norms

In first-order matrix optimization, as encountered in LLM training, "Fanions" denote a family of update methods parameterized by the Ky Fan $k$-norm and its dual. For $W\in \mathbb R^{m\times n}$ with singular values $\sigma_1\ge \cdots\ge 0$, the Ky Fan $k$-norm is

$\|W\|_{(k)} = \sum_{i=1}^k \sigma_i(W)$

with dual

$$\|X\|_{(k)}^\dagger = \max\{\|X\|_2,\, \tfrac{1}{k}\|X\|_*\} = \max\{\sigma_1(X),\, \tfrac{1}{k}\sum_{i=1}^r \sigma_i(X)\},\quad r=\min\{m,n\}$$

(Kravatskiy et al., 10 Dec 2025). Fanion-$k$ updates operate by taking the rank-$k$ projector onto the top singular directions of the gradient.

Convex combinations with the Frobenius norm (F-Fanions) or $\ell_\infty$ norm (S-Fanions) yield flexible update rules:

• F-Fanions mix the Ky Fan projector with normalized gradient;
• S-Fanions mix with a scaled elementwise sign. Notable extremes are F-Muon ($k=r$) and F-Neon ($k=1$), with Fanion-$r$ coinciding with the Muon update.

5. Muon-like Update Rules and LMO Implementation

Fanion updates are implemented within the Linear Minimization Oracle (LMO) framework. For a matrix $W$ and gradient $G$, each update is

$$D = \arg\min_{D:\,\|D\|\le1} \langle G, D \rangle, \qquad W \leftarrow W - \gamma D$$

where $\|\cdot\|$ is the dual norm described above. When specializing to the Fanion family:

• Fanion-$k$: $D = \sum_{i=1}^k u_iv_i^\top$ (top-$k$ singular directions)
• F-Muon: $$D = \alpha (\sum_{i=1}^r u_iv_i^\top) + (1-\alpha) G/\|G\|_F$$
• S-Muon: $$D = \alpha (\sum_{i=1}^r u_iv_i^\top) + (1-\alpha)\eta\,\mathrm{sign}(G)$$

Computational bottleneck is in computing the top-$k$ SVD, for which thick-restart Lanczos (TRLan) is efficient, with memory $O(mnd)$ and flops $O(mn^2 + n^2d + nd^2)$ when $m\ge n$ (Kravatskiy et al., 10 Dec 2025).

6. Theoretical Guarantees and Empirical Results

Standard convergence results for LMO-based algorithms extend to Fanions and their variants. For $L$–smooth objective $F$,

$\min_{0\le t<T}\mathbb E[\|\nabla F(W^t)\|_F^2] \le O\left(\frac{F(W^0) - F_\inf}{\gamma T}\right)$

with $\gamma=O(1/L\rho)$ if the dual ball is contained in a Frobenius ball of radius $\rho$, requiring $T=O(1/\epsilon^4)$ iterations to reach $\epsilon$–stationarity.

In convex settings with linear regression $F(W)=\frac{1}{2}\|AW-B\|_F^2$, one obtains linear convergence for all Fanion variants, with per-iteration scaling $O(\kappa^2\log(1/\epsilon))$ in the condition number $\kappa$.

Empirical findings demonstrate:\newline — F-Muon outperforms vanilla Muon (by 14%) on linear least squares;\newline — On CIFAR-10, NanoGPT, and GPT-2 Medium, F-Muon and S-Muon match or slightly outperform Muon;\newline — S-Muon and F-Muon remain robust across learning rates and problem settings, attesting to the flexibility provided by norm combinations (Kravatskiy et al., 10 Dec 2025).

Table: Fanion Algorithm Family

Name	Update Constraint	Interpolates with
Fanion-$k$	Ky Fan $k$-norm dual	Neon ($k=1$)
F-Muon	Ky Fan $r$ + Frobenius	NSGD (Frobenius)
S-Muon	Ky Fan $r$ + $\ell_\infty$	SignSGD

This explicit generalization unifies previous Muon, Neon, Dion, and normalized gradient methods under a shared structure (Kravatskiy et al., 10 Dec 2025).

7. Relationships, Practical Implications, and Future Directions

Fanions, in both tableau-combinatorial and matrix-optimization domains, possess deep structural connections. In optimization, the flexibility of choosing the norm constraint enables interpolation between low-rank, full-rank, and sign-based updates. Low-rank Fanion-$k$ variants provide fine-grained rank adaptivity, revealing failure modes of under-parameterized updates, such as Neon with $k=1$, versus robust performance at higher $k$.

Fanion methods support fast, scalable implementations and can be further enhanced with error feedback, adaptive steps, and layerwise dropout. The structural understanding of dual balls for composite norms underpins future algorithmic research, especially in large-scale, nonconvex deep learning regimes.

In algebraic combinatorics, flagged Schur functions enumerate and classify objects indexed by parabolic Catalan numbers, key tableaux, and Demazure characters, providing refined symmetry and convex-geometric insights for tableaux and their applications in representation theory (Proctor et al., 2016).