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The $q$-Schur algebras in type $D$, I: fundamental multiplication formulas

Jie Du, Yiqiang Li, Zhaozhao Zhao

TL;DR

The paper develops a robust framework for q-Schur algebras of type D by linking them to type B via an unequal-parameter Hecke algebra, and by establishing both algebraic and geometric realizations. It constructs two parallel theories—type B with unequal parameters and type D via restriction to the type D Hecke algebra—then uses orbit splitting and weight idempotents to transfer fundamental multiplication formulas from type B to type D. Central contributions include the algebraic definition of S_q^D(n,r) and its natural basis, a geometric realization on isotropic flag varieties, a detailed orbit-parametrization for SO and O groups, and a comprehensive suite of multiplication formulas (spanning several cases and subcases) that complete earlier gaps in FL and LL. These results lay the groundwork for a Schur–Weyl–Hecke duality in type D and provide explicit tools for studying representations of q-Schur algebras tied to orthogonal groups. The work thus advances both the structural understanding and computational toolkit for type D quantum symmetries and their geometric underpinnings.

Abstract

By embedding the Hecke algebra $\check H_q$ of type $D$ into the Hecke algebra $H_{q,1}$ of type $B$ with unequal parameters $(q,1)$, the $q$-Schur algebras $S^κ_q(n,r)$ of type $D$ is naturally defined as the endomorphism algebra of the tensor space with the $\check H_q$-action restricted from the $H_{q,1}$-action that defines the $(q,1)$-Schur algebra $S^\jmath_{q,1}(n,r)$ of type $B$. We investigate the algebras $S^\jmath_{q,1}(n,r)$ and $S^κ_q(n,r)$ both algebraically and geometrically and describe their standard bases, dimension formulas and weight idempotents. Most importantly, we use the geometrically derived two sets of the fundamental multiplication formulas in $S^\jmath_{q,1}(n,r)$ to derive multi-sets (9 sets in total!) of the fundamental multiplication formulas in $S^κ_q(n,r)$.

The $q$-Schur algebras in type $D$, I: fundamental multiplication formulas

TL;DR

The paper develops a robust framework for q-Schur algebras of type D by linking them to type B via an unequal-parameter Hecke algebra, and by establishing both algebraic and geometric realizations. It constructs two parallel theories—type B with unequal parameters and type D via restriction to the type D Hecke algebra—then uses orbit splitting and weight idempotents to transfer fundamental multiplication formulas from type B to type D. Central contributions include the algebraic definition of S_q^D(n,r) and its natural basis, a geometric realization on isotropic flag varieties, a detailed orbit-parametrization for SO and O groups, and a comprehensive suite of multiplication formulas (spanning several cases and subcases) that complete earlier gaps in FL and LL. These results lay the groundwork for a Schur–Weyl–Hecke duality in type D and provide explicit tools for studying representations of q-Schur algebras tied to orthogonal groups. The work thus advances both the structural understanding and computational toolkit for type D quantum symmetries and their geometric underpinnings.

Abstract

By embedding the Hecke algebra of type into the Hecke algebra of type with unequal parameters , the -Schur algebras of type is naturally defined as the endomorphism algebra of the tensor space with the -action restricted from the -action that defines the -Schur algebra of type . We investigate the algebras and both algebraically and geometrically and describe their standard bases, dimension formulas and weight idempotents. Most importantly, we use the geometrically derived two sets of the fundamental multiplication formulas in to derive multi-sets (9 sets in total!) of the fundamental multiplication formulas in .
Paper Structure (27 sections, 34 theorems, 187 equations)

This paper contains 27 sections, 34 theorems, 187 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The ${\boldsymbol q}$-Schur algebras of type $D$: an algebraic setting
  4. The Weyl groups $W/{\check W}$, Hecke algebras $H_{{\boldsymbol q},1}/\check H_{\boldsymbol q}$, and ${\boldsymbol q}$-Schur algebras $S^\jmath_{{\boldsymbol q},1}(n,r)/S^\kappa_{\boldsymbol q}(n,r)$
  5. Identifying $W,W_\lambda$ with certain fixed-point subgroups of ${\mathfrak S}_{2r}$
  6. Parabolic subgroups of ${\check W}$ as stabilizers of tabloids associated with signed compositions
  7. The ${\boldsymbol q}$-Schur algebras of type $D$
  8. A geometric setting for the ${\boldsymbol q}$-Schur algebras $S_{{\boldsymbol q},1}^\textsc{b}(n,r)$
  9. A geometric setting for the endomorphism algebra of a permutation module
  10. Finite orthogonal groups and flag varieties
  11. Parabolic subgroups of ${\text{\rm O}}_{2r}(q)$
  12. Parametrizing ${\mathrm{O}}_{2r}(q)$-orbits
  13. A geometric setting for $H^\textsc{b}_{{\boldsymbol q},1}$
  14. A geometric setting for $S^\textsc{b}_{{\boldsymbol q},1}(n,r)$
  15. A natural basis and geometric setting for $S^\textsc{d}_{\boldsymbol q}(n,r)$
  16. ...and 12 more sections

Key Result

Theorem 1.1

Let $S_{{\boldsymbol q},1}^\jmath(n,r)$ and $S_{{\boldsymbol q}}^\kappa(n,r)$ be the ${\boldsymbol q}$-Schur algebras of type $B$ and $D$, respectively, over the polynomial ring $\mathbb Z[{\boldsymbol q}]$ with their respective bases $\{e_A\mid A \in \Xi\}$ and $\{\phi_{\check{\mathbb A}}\mid {\ch for some $g_{h,A,p},g'_{_{h, A,\overline{p}}}\in\mathbb Z[{\boldsymbol q}]$. (See hp for the notat

Theorems & Definitions (69)

  • Theorem 1.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • Corollary 2.7
  • ...and 59 more