{
  "id": "2012.08411",
  "version": "v2",
  "published": "2020-12-15T16:44:23.000Z",
  "updated": "2020-12-31T10:39:08.000Z",
  "title": "Splitting Subspaces of Linear Operators over Finite Fields",
  "authors": [
    "Divya Aggarwal",
    "Samrith Ram"
  ],
  "comment": "13 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $V$ be a vector space of dimension $N$ over the finite field $\\mathbb{F}_q$ and $T$ be a linear operator on $V$. Given an integer $m$ that divides $N$, an $m$-dimensional subspace $W$ of $V$ is $T$-splitting if $V=W\\oplus TW\\oplus \\cdots \\oplus T^{d-1}W$ where $d=N/m$. Let $\\sigma(m,d;T)$ denote the number of $m$-dimensional $T$-splitting subspaces. Determining $\\sigma(m,d;T)$ for an arbitrary operator $T$ is an open problem. We prove that $\\sigma(m,d;T)$ depends only on the similarity class type of $T$ and give an explicit formula in the special case where $T$ is cyclic and nilpotent. We also show that $\\sigma(m,d;T)$ is a polynomial in $q$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2020-12-31T10:39:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite field",
      "linear operator",
      "splitting subspaces",
      "similarity class type",
      "vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}