• PDF / 974,375 Bytes
• 16 Pages / 439.642 x 666.49 pts Page_size
• 33 Downloads / 176 Views

DOWNLOAD

REPORT

---

Syzygies of Determinantal Thickenings and Representations of the General Linear Lie Superalgebra Claudiu Raicu1,2 · Jerzy Weyman3 Received: 7 March 2018 / Revised: 29 June 2018 / Accepted: 2 July 2018 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Abstract We let S = C[xi,j ] denote the ring of polynomial functions on the space of m × n matrices and consider the action of the group GL = GLm × GLn via row and column operations on the matrix entries. For a GL-invariant ideal I ⊆ S, we show that the linear strands of its minimal free resolution translate via the BGG correspondence to modules over the general linear Lie superalgebra gl(m|n). When I = Iλ is the ideal generated by the GL-orbit of a highest weight vector of weight λ, we give a conjectural description of the classes of these gl(m|n)-modules in the Grothendieck group, and prove that our prediction is correct for the first strand of the minimal free resolution. Keywords Determinantal thickenings · Syzygies · BGG correspondence · General linear Lie superalgebra · Kac modules · Dyck paths Mathematics Subject Classification (2010) Primary 13D02 · 14M12 · 17B10

1 Introduction We consider the vector space Cm×n of m × n complex matrices (m ≥ n) and let S = C[xi,j ] denote its coordinate ring. The group GL = GLm (C) × GLn (C) acts on Cm×n via  Claudiu Raicu

[email protected] Jerzy Weyman [email protected] 1

Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, USA

2

Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania

3

Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

C. Raicu and J. Weyman

row and column operations, making S into a GL-representation whose decomposition into irreducible representations is governed by Cauchy’s formula: if we write Nndom for the set of partitions with at most n parts (i.e., dominant weights in Zn with non-negative entries) and write Sλ for the Schur functor associated to a partition λ then we have using [15, Corollary 2.3.3] that S=



Sλ Cm ⊗ Sλ Cn .

(1.1)

λ∈Nndom

When I ⊆ S is a GL-invariant ideal, the syzygy modules TorSi (I, C) are naturally representations of GL, but their explicit description is known only in special cases [1, 5, 6, 9]. By contrast, ExtiS (I, S) can be described for every GL-invariant ideal I ⊆ S as explained in [7]. A special class of GL-invariant ideals consists of the ones generated by a single summand Sλ Cm ⊗Sλ Cn in (1.1), and are denoted by Iλ : one can think of them as principal GL-invariant ideals, in the sense that they are generated by the GL-orbit of a single highest weight vector. The goals of this article are to propose a conjectural description of TorSi (Iλ , C) for an arbitrary partition λ and to give supporting evidence for our conjecture. To formulate our conjecture, we re-express the problem of computing syzygies into one about modules over the exterior algebra via the BGG correspondence (described in Section 2.