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+================================================================
+PARABOLIC INTERSECTIONS IN ARTIN GROUPS
+A gentle introduction
+================================================================
+
+This document explains a recent theorem about a family of groups
+called ARTIN GROUPS. It is written for a reader who knows what a
+group is (or is willing to learn in Section 1), likes concrete
+examples, and wants to understand what the theorem claims and
+roughly why it is true. The full research proof is 27,000 words
+of graduate-level topology; we will state the theorem precisely,
+explain the proof's big idea, and hand-verify small examples.
+
+Honesty note up front: some inner steps of the proof need linear
+algebra, topology of chain complexes, and a bit of fusion theory.
+We will be honest about which parts are accessible and which are
+"trust the specialists, here is the shape of their argument."
+The theorem itself is a yes/no statement about subgroups, and
+that statement is completely accessible.
+
+================================================================
+1. WHAT IS A GROUP
+================================================================
+
+A GROUP is a set G together with an operation * (think "combine
+two things") satisfying:
+
+ 1. Closure: if a, b are in G, then a*b is in G.
+ 2. Associativity: (a*b)*c = a*(b*c).
+ 3. Identity: there is an element 1 with 1*a = a*1 = a.
+ 4. Inverse: every a has an a^{-1} with a*a^{-1} = 1.
+
+Examples:
+ - The integers under addition. Identity 0. Inverse -n.
+ - The integers mod 7 under addition.
+ - The nonzero rationals under multiplication.
+ - Rotations of a square: 0, 90, 180, 270 degrees. 4 elements.
+ - All permutations of {1, 2, 3}: the symmetric group S_3, with
+ 6 elements.
+
+A PRESENTATION writes a group as <generators | relations>. The
+group is "the largest group in which those relations hold". Two
+words represent the same element iff you can convert one to the
+other using the relations plus trivial bookkeeping.
+
+ Z (integers) = <a | >. No relations. a^n means "+n".
+ Z/7Z = <a | a^7 = 1>.
+ Z^2 = <a, b | ab = ba>. Pairs of integers, added pointwise.
+ F_2 (free group on 2 letters) = <a, b | >. No relations.
+
+================================================================
+2. COXETER MATRICES
+================================================================
+
+A COXETER MATRIX on a finite set S is a symmetric table m(s,t)
+of "braid lengths":
+
+ m(s, s) = 1
+ m(s, t) = m(t, s)
+ m(s, t) in {2, 3, 4, 5, ...} or the symbol "infinity"
+
+Intuition: m(s,t) tells you how strongly s and t interact.
+
+ m = 2: s and t commute (weakest interaction)
+ m = 3: s t s = t s t (the "braid" relation)
+ m = 4: s t s t = t s t s (dihedral relation of order 4)
+ m = infinity: no relation at all between s and t
+
+Draw a picture: one node per element of S, one edge labelled m(s,t)
+between s and t whenever m(s,t) >= 3. Nodes with no edge commute.
+Nodes joined by an edge labelled "infinity" do not interact.
+
+Examples:
+ - S = {a, b}, m(a, b) = 2: two commuting nodes.
+ - S = {a, b}, m(a, b) = 3: a "braided" pair.
+ - S = {a, b}, m(a, b) = infinity: a "free" pair.
+ - S = {a, b, c}, m(a, b) = m(b, c) = 3, all other pairs 2:
+ the "A_3 diagram" (three nodes in a line with labelled edges;
+ a and c commute because m(a, c) = 2).
+
+A Coxeter matrix is SPHERICAL if its associated Coxeter group
+(defined in Section 3) is finite. Spherical diagrams have been
+classified since the 1930's: types A_n, B_n, D_n, E_6, E_7, E_8,
+F_4, G_2, H_3, H_4, I_2(m), and disjoint unions.
+
+================================================================
+3. ARTIN GROUPS AND COXETER GROUPS
+================================================================
+
+Fix a Coxeter matrix m on finite set S. The ARTIN GROUP A = A_M
+is the group with generators s (for s in S) and one relation per
+finite edge:
+
+ for m(s, t) < infinity:
+ sts... (m letters) = tst... (m letters)
+
+The COXETER GROUP W = W_M is what you get by ALSO imposing s^2 = 1
+for every s.
+
+Examples:
+
+ All m(s, t) = 2 (zero edges):
+ A_M = Z^n (free abelian group on n generators).
+ W_M = (Z/2)^n.
+
+ All m(s, t) = 3, on n generators in a line (the "A_n" diagram):
+ A_M = Braid_{n+1} (braid group on n+1 strands).
+ W_M = S_{n+1} (symmetric group on n+1 letters).
+
+ All m(s, t) = infinity (no relations beyond nothing):
+ A_M = F_n (free group on n generators).
+ W_M = (Z/2) * (Z/2) * ... (free product of n copies of Z/2).
+
+ S = {a}, single generator:
+ A_M = Z, W_M = Z/2.
+
+ S = {a, b}, m(a, b) = 3:
+ A_M = Braid_3 = <a, b | aba = bab>.
+ W_M = S_3 (order 6).
+
+ S = {a, b}, m(a, b) = 4:
+ A_M = <a, b | abab = baba> (the "I_2(4)" Artin group).
+ W_M = the dihedral group of order 8 (symmetry of a square).
+
+ S = {a, b, c}, m(a, b) = m(b, c) = 2, m(a, c) = infinity:
+ A_M = <a, b, c | ab = ba, bc = cb> (RIGHT-ANGLED Artin group
+ on the path a - b - c).
+ W_M = (Z/2 x Z/2) *_<b> (Z/2 x Z/2), infinite.
+ A_M is nonspherical.
+
+Braid group on 3 strands, Braid_3 = <a, b | aba = bab>, is the
+smallest interesting spherical example. It is infinite, so an
+Artin group need not be finite even when it is "spherical type".
+
+================================================================
+4. PARABOLIC SUBGROUPS
+================================================================
+
+Fix a Coxeter matrix m on S, Artin group A = A_M.
+
+For any subset X of S, let A_X be the subgroup generated by
+{s : s in X}. (It is itself an Artin group, on the restricted
+matrix.) Call A_X a STANDARD PARABOLIC subgroup of rank |X|.
+
+A PARABOLIC subgroup is any conjugate of a standard parabolic:
+
+ g A_X g^{-1}, where X is a subset of S and g is in A.
+
+Two notable things:
+
+ - Different conjugates of the same A_X can be different
+ subgroups. For instance in F_2 = <a, b>, <a> and <b a b^{-1}>
+ are two different infinite cyclic subgroups, both conjugates
+ of <a>. Their intersection is {1}.
+
+ - The word "parabolic" comes from the geometry of hyperplane
+ arrangements. Coxeter groups act on Euclidean spaces by
+ reflections; the "parabolic" subgroups are stabilizers of
+ points where certain coordinate directions have collapsed.
+ The name is inherited by the group-theoretic definition.
+
+Examples:
+ A = Z^2: standard parabolics are {0}, Z*a, Z*b, Z^2.
+ Since A is abelian, every parabolic is standard.
+ A = F_2: standard parabolics are {1}, <a>, <b>, F_2.
+ Conjugates of <a> are the infinite cyclic subgroups
+ <w a w^{-1}> for w in F_2.
+ A = Braid_3: parabolics are {1}, and conjugates of <a>, <b>,
+ and Braid_3 itself.
+
+================================================================
+5. THE PARABOLIC INTERSECTION CONJECTURE
+================================================================
+
+Informally: intersection of any two parabolics is again parabolic.
+
+Precisely, the theorem of this document:
+
+THEOREM (Main). For every Coxeter matrix M on a finite set S,
+every subsets X, Y of S, and every elements g, h in A, there are
+a subset Z of S and an element k in A such that
+
+ g A_X g^{-1} INTERSECT h A_Y h^{-1} = k A_Z k^{-1}.
+
+In words: intersect two conjugates of standard parabolics; the
+result is another conjugate of a standard parabolic.
+
+That is the whole theorem. Two features make it non-trivial:
+
+ (1) The ambient Artin group A is allowed to be nonspherical.
+ Many classical tools (Garside structure, normal forms via
+ the "positive monoid", parabolic closures computed from
+ Garside theory) assume sphericity or at least "FC type"
+ (every pair of generators with finite label forms a
+ spherical pair). The theorem does not.
+
+ (2) The Coxeter matrix M can have infinite labels. Then
+ positive monoids fail to admit normal forms in the
+ classical way; the "positive fractional" machinery breaks.
+
+================================================================
+6. WHAT WAS KNOWN BEFORE 2026
+================================================================
+
+Cases:
+ 1997 Paris, 1970's van der Lek: A_X ∩ A_Y = A_{X ∩ Y} for
+ standards (parabolics generated by subsets). This is easy.
+ 2007 Duncan-Kazachkov-Remeslennikov: arbitrary intersections of
+ parabolics are parabolic, for RIGHT-ANGLED Artin groups.
+ 2019 Cumplido-Gebhardt-Gonzalez-Meneses-Wiest: the conjecture
+ for SPHERICAL Artin groups (using Garside theory).
+ 2021 Morris-Wright: FC-type, one side spherical.
+ 2022 Antolin-Foniqi: even FC-type, both sides spherical.
+ 2023 Cumplido-Martin-Vaskou: LARGE type.
+ 2024 Moller-Paris-Varghese: one side spherical, arbitrary
+ ambient. Also extended pairwise to arbitrary intersections,
+ ASSUMING pairwise.
+ 2026 Character (this paper): full case, arbitrary ambient,
+ arbitrary labels.
+
+================================================================
+7. THE STRATEGY: REPRESENT A ON SOMETHING
+================================================================
+
+Intersections of subgroups are hard to see from a presentation.
+The standard trick is to REPRESENT the group: find some object on
+which A acts, in such a way that subgroups correspond to "things
+that are fixed".
+
+Ordinary example: S_3 acts on a triangle. The subgroup fixing
+vertex #1 is a parabolic (a stabilizer of a point).
+
+For Artin groups, the right representation is on a CATEGORY of
+CHAIN COMPLEXES over a finite auxiliary algebra Z. The paper
+builds Z explicitly, defines an action of A on these complexes,
+and then translates:
+
+ "a is in A_X" becomes "P_{f, L} is preserved by a, for
+ all labels L"
+
+for a distinguished projective P_{f, L} built inside Z-complexes.
+
+Rough plan:
+ STEP 1 (Section 8): Build Z and an action of A by invertible
+ functors on the homotopy category of graded
+ Z-complexes.
+
+ STEP 2 (Section 9): Exhibit "test" objects P_{f, L} that
+ satisfy: for every x in A, the set
+ {a in A : B_a P_{f, L}(x) is homotopy
+ equivalent to P_{f, L}(x) for all L}
+ equals the stabilizer H(S, x), and
+ H(S, 1) = A_X.
+
+ STEP 3 (Section 10-13): Prove the STABILIZER THEOREM: for any
+ subset Y and any x in A, the stabilizer
+ H(Y, x) is itself a conjugate of a
+ standard parabolic of A_Y.
+
+ STEP 4 (Section 14): Use the case Y = S to conclude the main
+ theorem.
+
+================================================================
+8. THE AUXILIARY ALGEBRA Z
+================================================================
+
+We need a finite-dimensional graded algebra Z. (Graded means: Z
+splits as a direct sum of pieces Z^0, Z^1, Z^2, ...; multiplying
+an element of Z^p by an element of Z^q gives an element of
+Z^{p+q}. For us Z is finite-dimensional over a field, and
+everything lives in a bounded range of degrees.)
+
+Construction idea (approximate, hand-wavey):
+
+ - Attach a special vertex f to the diagram.
+ - The vertex f interacts with each generator s of S. The
+ interaction is encoded by a "fusion rule" with a small tensor
+ category.
+ - Z = the path algebra of a quiver whose vertices are "colour
+ and label" positions and whose arrows implement the labelling.
+
+The important consequence is a certain TWIST MECHANISM: for each
+generator s of A, there is a bimodule chain complex B_s such
+that:
+
+ (i) B_s is invertible up to chain homotopy equivalence.
+ (ii) B_s B_t B_s = B_t B_s B_t (chain homotopy) whenever
+ m(s, t) = 3.
+ (iii) Similarly for m(s, t) = 4, 5, ..., infinity (the "no
+ relation" case means the twists B_s and B_t commute up to
+ homotopy, which is exactly the right-angled paradigm
+ m(s, t) = 2 generalised).
+
+So we get an honest action of A on the homotopy category of
+bounded graded Z-complexes. For a word x in A, B_x is the
+corresponding functor.
+
+The essential trick: the action of B_s on a complex is by POSITIVE
+operators: differentials never decrease a certain degree called
+"level" (Section 9), and B_s increases the level by at most 1.
+The B_s^{-1} decreases by at most 1.
+
+================================================================
+9. LAYERS, TESTS, AND HEIGHT
+================================================================
+
+A generator of a complex has two degrees: COCHAIN degree (call it
+c) and INTERNAL degree (call it u). The LEVEL is d = c + u.
+This bookkeeping accounts for all shifts.
+
+For a complex E, the LOW subcomplex t_{<=k} E has all generators
+with level <= k. The HIGH quotient t_{>=k+1} E has all generators
+with level >= k+1. The short exact sequence
+
+ 0 -> t_{<=k} E -> E -> t_{>=k+1} E -> 0
+
+is called the TRUNCATION TRIANGLE at level k.
+
+The PURE LAYER at level d, written Y^d(E), is the subquotient
+t_{<=d} / t_{<=d-1}: the generators at level exactly d, with their
+differentials.
+
+Because every B_s has amplitude [0, 1] (bounded level change), the
+layers of B_s E are controlled by the layers of E. In particular,
+for a pure input at level d, B_s E has just two layers: a "bottom"
+at d and an "upper" at d+1 (upper supported only on colour s).
+
+The KEY INSTRUMENT: for a vertex a (corresponding to a generator s
+and a label L), the RIGHT TEST of E at a is:
+
+ e_a E = (right projective at a) TENSOR E
+
+It is a complex of plain vector spaces. Its cohomology splits by
+ordinary cochain degree. Two specific degrees reveal the layer:
+
+ H^d(e_a E) = socle of Y^d(E) at a
+ H^{d+2}(e_a E) = (top of Y^d(E) at a) <2> (internal shift 2)
+
+So the "degree-d" and "degree-(d+2)" tests pick out the socle and
+top of the pure layer. In particular a layer is nonzero at colour
+s iff one of these special cohomology classes is nonzero.
+
+HEIGHT. For a word x in A, the FAMILY at x is:
+
+ E_L(x) = B_x P_{f, L} for each label L.
+
+The (relative) HEIGHT of a word a in A_Y relative to x is the
+maximum total degree (cochain + internal) at which ANY test of ANY
+member of E(x) is nonzero. Finite whenever nonempty. The height
+is:
+
+ - nondecreasing under positive words in A_Y (Section 13);
+ - strictly increasing under any positive word that contains
+ every colour of Y (Lemma 7.cochain-drop of the paper).
+
+The height being bounded along a sequence of positive words forces
+strong consequences. That is the central engine of the proof.
+
+-- Section 10 continues in the next chunk of this file --