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We give a necessary and sufficient condition for two real flag manifolds, which are not necessarily congruent, in a complex flag manifold to intersect transversally in terms of the symmetric triad. Then we show that the intersection of two real flag manifolds is antipodal. As an application, we prove that any real flag manifold in a complex flag manifold is a globally tight Lagrangian submanifold.

By identifying each standard flag with a trivalent Feynman diagram, the corresponding propagators can be read directly from the flag itself. Within the flag representation, the kinematic Jacobi identity (equivalently, the residue theorem on moduli spaces) admits a natural interpretation as the equivalence between a complete flag and its gapped counterpart. Using flags together with Orlik-Solomon algebras, we reconstruct the intersection numbers of twisted cocycles, thereby obtaining the bi-adjoint amplitude. Moreover, employing flag simplices enables the construction of the Z-amplitude in the alpha' -> 0 limit. By further examining pairings of specific flags, we also recover the Cachazo-He-Yuan (CHY) representation of the bi-adjoint amplitude.

We provide a new axiom system for flag matroids, characterize representability of uniform flag matroids, and give forbidden minor characterizations of full flag matroids that are representable over $\mathbb{F}_2$ and $\mathbb{F}_3$ along with regular full flag matroids. We also provide different equivalent characterizations for regular full flag matroids.

Given a finite field F_q and a positive integer n, a flag is a sequence of nested F_q-subspaces of a vector space F_q^n and a flag code is a nonempty collection of flags. The projected codes of a flag code are the constant dimension codes containing all the subspaces of prescribed dimensions that form the flags in the flag code. In this paper we address the notion of equivalence for flag codes and explore in which situations such an equivalence can be reduced to the equivalence of the corresponding projected codes. In addition, this study leads to new results concerning the automorphism group of certain families of flag codes, some of them also introduced in this paper.

We study the totally nonnegative part of the complete flag variety and of its tropicalization. We show that Lusztig's notion of nonnegative complete flag variety coincides with the flags in the complete flag variety which have nonnegative Pl{ü}cker coordinates. This mirrors the characterization of the totally nonnegative Grassmannian as those points in the Grassmannian whose Pl{ü}cker coordinates are all nonnegative. We then study the tropical complete flag variety and complete flag Dressian, which are two tropical versions of the complete flag variety, capturing realizable and abstract flags of tropical linear spaces, respectively. In general, the complete flag Dressian properly contains the tropical complete flag variety. However, we show that the totally nonnegative parts of these spaces coincide.

In network coding, a flag code is a set of sequences of nested subspaces of $\mathbb{F}_q^n$, being $\mathbb{F}_q$ the finite field with $q$ elements. Flag codes defined as orbits of a cyclic subgroup of the general linear group acting on flags of $\mathbb{F}_q^n$ are called cyclic orbit flag codes. Inspired by the ideas in arXiv:1403.1218, we determine the cardinality of a cyclic orbit flag code and provide bounds for its distance with the help of the largest subfield over which all the subspaces of a flag are vector spaces (the best friend of the flag). Special attention is paid to two specific families of cyclic orbit flag codes attaining the extreme possible values of the distance: Galois cyclic orbit flag codes and optimum distance cyclic orbit flag codes. We study in detail both classes of codes and analyze the parameters of the respective subcodes that still have a cyclic orbital structure.

We study the totally non-negative part of the complete flag variety and of its tropicalization. We start by showing that Lusztig's notion of non-negative complete flag variety coincides with the flags in the complete flag variety which have non-negative Plücker coordinates. This mirrors the characterization of the totally non-negative Grassmannian as those points in the Grassmannian with all non-negative Plücker coordinates. We then study the tropical complete flag variety and complete flag Dressian, which are two tropical versions of the complete flag variety, capturing realizable and abstract flags of tropical linear spaces, respectively. The complete flag Dressian properly contains the tropical complete flag variety. However, we show that the totally non-negative parts of these spaces coincide.

A flag positroid of ranks $\boldsymbol{r}:=(r_1<\dots <r_k)$ on $[n]$ is a flag matroid that can be realized by a real $r_k \times n$ matrix $A$ such that the $r_i \times r_i$ minors of $A$ involving rows $1,2,\dots,r_i$ are nonnegative for all $1\leq i \leq k$. In this paper we explore the polyhedral and tropical geometry of flag positroids, particularly when $\boldsymbol{r}:=(a, a+1,\dots,b)$ is a sequence of consecutive numbers. In this case we show that the nonnegative tropical flag variety TrFl$_{\boldsymbol{r},n}^{\geq 0}$ equals the nonnegative flag Dressian FlDr$_{\boldsymbol{r},n}^{\geq 0}$, and that the points $\boldsymbolμ = (μ_a,\ldots, μ_b)$ of TrFl$_{\boldsymbol{r},n}^{\geq 0} =$ FlDr$_{\boldsymbol{r},n}^{\geq 0}$ give rise to coherent subdivisions of the flag positroid polytope $P(\underline{\boldsymbolμ})$ into flag positroid polytopes. Our results have applications to Bruhat interval polytopes: for example, we show that a complete flag matroid polytope is a Bruhat interval polytope if and only if its $(\leq 2)$-dimensional faces are Bruhat interval polytopes. Our results also have applications to realizability questions. We define a positively oriented flag m

The flag curvature is a natural Finsler extension of the sectional curvature in Riemannian geometry. However, there are many non-Riemannian quantities which interact with the flag curvature. In this paper, we introduce a notion of weighted flag curvature by modifying the flag curvature using the non-Riemannian quantity, $T$-curvature. We show that a forward complete open Finsler manifold with positive weighted flag curvature is necessarily diffeomorphic to the Euclidean space, and that a compact Finsler manifold with nonnegative weighted flag curvature and strictly convex boundary is diffeomorphic to a Euclidean ball.

Topological data analysis (TDA) has had enormous success in science and engineering in the past decade. Persistent topological Laplacians (PTLs) overcome some limitations of persistent homology, a key technique in TDA, and provide substantial insight to the behavior of various geometric and topological objects. This work extends PTLs to directed flag complexes, which are an exciting generalization to flag complexes, also known as clique complexes, that arise naturally in many situations. We introduce the directed flag Laplacian and show that the proposed persistent directed flag Laplacian (PDFL) is a distinct way of analyzing these flag complexes. Example calculations are provided to demonstrate the potential of the proposed PDFL in real world applications.

Flag matroids are combinatorial abstractions of flags of linear subspaces, just as matroids are of linear subspaces. We introduce the flag Dressian as a tropical analogue of the partial flag variety, and prove a correspondence between: (a) points on the flag Dressian, (b) valuated flag matroids, (c) flags of projective tropical linear spaces, and (d) coherent flag matroidal subdivisions. We introduce and characterize projective tropical linear spaces, which serve as a fundamental tool in our proof. We apply the correspondence to prove that all valuated flag matroids on ground set up to size 5 are realizable, and give an example where this fails for a flag matroid on 6 elements.

We study linear degenerations of flag varieties from the point of view of tropical geometry. We define the linear degenerate flag Dressian and prove a correspondence between: $(a)$ points in the linear degenerate flag Dressian, $(b)$ linear degenerate flags of valuated matroids, $(c)$ linear degenerate flags of tropical linear spaces, and $(d)$ morphisms of valuated matroids arising from projection maps.

Motzkin numbers have been widely studied since they count many different combinatorial objects. In this paper we present a new appearance of this remarkable sequence in the network coding setting through a particular case of multishot codes called flag codes. A flag code is a set of sequences of nested subspaces (flags) of a vector space over the finite field $\mathbb{F}_q$. If the list of dimensions is $(1, \dots, n-1)$, we speak about a full flag code. The flag distance is defined as the sum of the respective subspace distances and can be represented by means of the so-called distance vectors. We show that the number of distance vectors corresponding to the full flag variety on $\mathbb{F}_q^n$ is exactly the $n$-th Motzkin number. Moreover, we can identify the integer sequence that counts the number of possible distance vectors associated to a full flag code with prescribed minimum distance.

Hands-on training sessions become a standard way to develop and increase knowledge in cybersecurity. As practical cybersecurity exercises are strongly process-oriented with knowledge-intensive processes, process mining techniques and models can help enhance learning analytics tools. The design of our open-source analytical dashboard is backed by guidelines for visualizing multivariate networks complemented with temporal views and clustering. The design aligns with the requirements for post-training analysis of a special subset of cybersecurity exercises -- supervised Capture the Flag games. Usability is demonstrated in a case study using trainees' engagement measurement to reveal potential flaws in training design or organization.

Consider a complete flag $\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$ of one-point AG codes of length $n$ over the finite field $\mathbb{F}$. The codes are defined by evaluating functions with poles at a given point $Q$ in points $P_1,\dots,P_n$ distinct from $Q$. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point $Q$, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In [3] it is shown, for a curve of genus $g$, that a flag of one-point AG codes defined with a subset of $n > 2g+2$ rational points is isometry-dual if and only if the last code $C_n$ in the flag is defined with functions of pole order at most $n+2g-1$. Using a different approach, we extend this characterization to all subsets of size $n \geq 2g+2$. Moreover we show that this is best possible by giving examples of isometry

This paper is a direct generalization of Baker-Bowler theory to flag matroids, including its moduli interpretation as developed by Baker and the second author for matroids. More explicitly, we extend the notion of flag matroids to flag matroids over any tract, provide cryptomorphic descriptions in terms of basis axioms (Grassmann-Plücker functions), circuit/vector axioms and dual pairs, including additional characterizations in the case of perfect tracts. We establish duality of flag matroids and construct minors. Based on the theory of ordered blue schemes, we introduce flag matroid bundles and construct their moduli space, which leads to algebro-geometric descriptions of duality and minors. Taking rational points recovers flag varieties in several geometric contexts: over (topological) fields, in tropical geometry, and as a generalization of the MacPhersonian.

In this paper we define and study a generalization of the Belinson-Drinfeld Grassmannian to the case where the curve is replaced by a smooth projective surface $X$, and the trivialization data are given on loci suitably associated to a nonlinear flag of closed subschemes. In order to do this, we first establish some general formal gluing results for moduli of almost perfect complexes, perfect complexes and torsors. We then construct a simplicial object $Fl_X$ of flags of closed subschemes of a smooth projective surface $X$, naturally associated to the operation of taking union of flags. We prove that this simplicial object has the 2-Segal property. For an affine complex algebraic group $G$, we finally define a derived, flag analog $Gr_X$ of the Beilinson-Drinfeld Grassmannian of $G$-bundles on the surface $X$, and show that most of the properties of the Beilinson-Drinfeld Grassmannian for curves can be extended to our flag generalization. In particular, we prove a factorization formula, the existence of a canonical flat connection, and define a chiral product on suitable sheaves on $Fl_X$ and on $Gr_X$. We also sketch the construction of actions of flags analogs of the loop group a

Flag codes that are orbits of a cyclic subgroup of the general linear group acting on flags of a vector space over a finite field, are called cyclic orbit flag codes. In this paper we present a new contribution to the study of such codes started in arXiv:2102.00867, by focusing this time on the generating flag. More precisely, we examine those ones whose generating flag has at least one subfield among its subspaces. In this situation, two important families arise: the already known Galois flag codes, in case we have just fields, or the generalized Galois flag codes in other case. We investigate the parameters and properties of the latter ones and explore the relationship with their underlying Galois flag code.

Given $\mathbb{F}_q$ the finite field with $q$ elements and an integer $n\geq 2$, a flag is a sequence of nested subspaces of $\mathbb{F}_q^n$ and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. Our study of the flag distance by using this new tool allows us to provide a fine description of the structure of flag codes as well as to derive bounds for their maximum possible size once the minimum distance and dimensions are fixed.