--- title: 'Fanions: Flagged Schur & Matrix Updates' url: https://www.emergentmind.com/topics/fanions type: topic --- # Fanions: Flagged Schur & Matrix Updates 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 $[i,\varphi_i]$ [1612.06323]. 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 $R=\{q_1<\dots\pi_b<\pi_c$ with $a\le q_h< b\le q_{h+1}\pi_c$. 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 [1612.06323]. ## 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 $\mathcal D_\lambda(\pi) = \{S\le Y_\lambda(\pi)\}$ [1612.06323]. 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 large language model (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 \[ \|X\|_{(k)}^\dagger = \max\{\|X\|_2,\, \tfrac{1}{k}\|X\|_*\} = \max\{\sigma_1(X),\, \tfrac{1}{k}\sum_{i=1}^r \sigma_i(X)\},\quad r=\min\{m,n\} \] [2512.09678]. 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 $m\ge n$ [2512.09678]. ## 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