{
  "id": "1809.03644",
  "version": "v1",
  "published": "2018-09-11T00:55:26.000Z",
  "updated": "2018-09-11T00:55:26.000Z",
  "title": "Pseudocharacters of Classical Groups",
  "authors": [
    "Matthew Weidner"
  ],
  "comment": "17 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "A $GL_d$-pseudocharacter is a function from a group $\\Gamma$ to a ring $k$ satisfying polynomial relations which make it \"look like\" the character of a representation. When $k$ is an algebraically closed field, Taylor proved that $GL_d$-pseudocharacters of $\\Gamma$ are the same as degree-$d$ characters of $\\Gamma$ with values in $k$, hence are in bijection with equivalence classes of semisimple representations $\\Gamma \\rightarrow GL_d(k)$. Recently, V. Lafforgue generalized this result by showing that, for any connected reductive group $H$ over an algebraically closed field $k$ of characteristic 0 and for any group $\\Gamma$, there exists an infinite collection of functions and relations which are naturally in bijection with $H^0(k)$-conjugacy classes of semisimple representations $\\Gamma \\rightarrow H(k)$. In this paper, we reformulate Lafforgue's result in terms of a new algebraic object called an FFG-algebra. We then define generating sets and generating relations for these objects and show that, for all $H$ as above, the corresponding FFG-algebra is finitely presented. Hence we can always define $H$-pseudocharacters consisting of finitely many functions satisfying finitely many relations. Next, we use invariant theory to give explicit finite presentations of the FFG-algebras for (general) orthogonal groups, (general) symplectic groups, and special orthogonal groups. Finally, we use our pseudocharacters to answer questions about conjugacy vs. element-conjugacy of representations, following Larsen.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-11T00:55:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G05",
      "13A50"
    ],
    "keywords": [
      "pseudocharacter",
      "classical groups",
      "algebraically closed field",
      "semisimple representations",
      "ffg-algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}