{
  "id": "2405.18592",
  "version": "v1",
  "published": "2024-05-28T21:10:40.000Z",
  "updated": "2024-05-28T21:10:40.000Z",
  "title": "Invariant Subspaces of Nilpotent Operators. Level, Mean, and Colevel: The Triangle $\\Bbb T(n)$",
  "authors": [
    "Claus Michael Ringel",
    "Markus Schmidmeier"
  ],
  "comment": "The manuscript has 137 illustrations",
  "categories": [
    "math.RT",
    "math.CO"
  ],
  "abstract": "We consider the category $\\mathcal S(n)$ of all pairs $X = (U,V)$, where $V$ is a finite-dimensional vector space with a nilpotent operator $T$ with $T^n = 0$, and $U$ is a subspace of $V$ such that $T(U) \\subseteq U$. Our main interest in an object $X=(U,V)$ are the three numbers $uX=\\dim U$ (for the subspace), $wX=\\dim V/U$ (for the factor) and $bX=\\dim {\\rm Ker} T$ (for the operator). Actually, instead of looking at the reference space $\\Bbb R^3$ with the triples $(uX,wX,bX)$, we will focus the attention to the corresponding projective space $\\Bbb T(n)$ which contains for a non-zero object $X$ the level-colevel pair {\\bf pr}$X = (uX/bX,wX/bX)$ supporting the object $X$. We use $\\Bbb T(n)$ to visualize part of the categorical structure of $\\mathcal S(n)$: The action of the duality $D$ and the square $\\tau_n^2$ of the Auslander-Reiten translation are represented on $\\Bbb T(n)$ by a reflection and a rotation by $120^\\circ$ degrees, respectively. Moreover for $n\\geq 6$, each component of the Auslander-Reiten quiver of $\\mathcal S(n)$ has support either contained in the center of $\\Bbb T(n)$ or with the center as its only accumulation point. We show that the only indecomposable objects $X$ in $\\mathcal S(n)$ with support having boundary distance smaller than 1 are objects with $bX=1$ which lie on the boundary, whereas any rational vector in $\\Bbb T(n)$ with boundary distance at least 2 supports infinitely many indecomposable objects. At present, it is not clear at all what happens for vectors with boundary distance between 1 and 2. The use of $\\Bbb T(n)$ provides even in the (quite well-understood) case $n = 6$ some surprises: In particular, we will show that any indecomposable object in $\\mathcal S(6)$ lies on one of 12 central lines in $\\Bbb T(6)$. The paper is essentially self-contained, all prerequisites which are needed are outlined in detail.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-28T21:10:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "16G70",
      "16E65",
      "47A15"
    ],
    "keywords": [
      "nilpotent operator",
      "invariant subspaces",
      "indecomposable object",
      "finite-dimensional vector space",
      "boundary distance smaller"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}