Abstract
We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson–Lusztig–MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated with any weight function. These bases are the counterparts of Lusztig’s bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson–Lusztig–MacPherson construction, which is first introduced by Fan–Li from a geometric viewpoint.
1 Introduction
1.1 Background
The quantum groups introduced by Drinfeld and Jimbo have played a central role in representation theory and many other branches of mathematics. Equally important are Lusztig’s modified (or idempotented) quantum groups (cf. [22]) that admit the canonical bases, which are analogs of the Kazhdan–Lusztig bases for the Hecke algebras. In [4], a geometric construction of the modified quantum group |$\dot{\textbf U} \ (\mathfrak{g}\mathfrak{l}_n)$| is given by Beilinson–Lusztig–MacPherson. Their construction is now referred as the BLM or stabilization construction after a stabilization property of the family of the (quantum) Schur algebras of type A. In this paper, by a (equal-parameter) stabilization construction of type |$X$| we mean a construction of an algebra |$\dot{\textbf{K}}^X_n$| over |${\mathbb{Z}}[v,v^{-1}]$| such that
(1) There is a family of quantum Schur algebras |$\textbf{S}^X_{n,d}$|, which are the centralizing algebras to the action of the Hecke algebra |$\textbf{H}^X_d$| of type |$X_d$|, for all |$n,d$|;
(2) The family |$\{\textbf{S}^X_{n,d} \ | \ d\in{\mathbb{N}}\}$| admits a stabilization property, namely, the algebra |$\dot{\textbf{K}}^X_n = \mathop{\textrm{Stab}}\limits _{\infty \leftarrow d} \textbf{S}^X_{n,d}$| is well defined. As a consequence, there is a basis of |$\dot{\textbf{K}}^X_n$| that is compatible with the Kazhdan–Lusztig bases for |$\textbf{H}^X_d$|, and the canonical bases of |$\textbf{S}^X_{n,d}$| for all |$d$|.
The stabilization constructions have been developed for classical type and for certain affine type (see Table 1 for the references)—there are geometric approaches using partial flags and counting over finite fields developed, while there also are algebraic approaches in the framework of the Hecke algebras using combinatorics on Coxeter groups.
We remark that the algebraic approach for finite type B/C is more or less a special case for affine type C, while the algebraic approach for type D will be given in the appendix of this present paper.
The stabilization construction in general produces not the Drinfeld–Jimbo’s quantum groups but Letzter–Kolb’s quantum symmetric pairs (cf. [18, 21]). For example, the stabilization constructions of type A and B/C lead to the quantum symmetric pairs of type AIII/IV with no black nodes.
1.2 A new direction
A recent work by Bao–Wang–Watanabe brings to the authors’ attention that a multiparameter Schur duality (cf. [7]) plays a governing role among the Schur dualities of classical type. They also introduce a multiparameter upgrade of quantum symmetric pairs of type AIII/AIV with no black nodes.
While it is unclear how to proceed a geometric approach with unequal parameters since dimension counting does not make sense in an obvious way, an algebraic/combinatorial approach seems viable. The goal of this article is to provide a stabilization construction with respect to the Schur duality with unequal parameters in loc. cit. We show that the multiparameter stabilization algebras constructed are the coideal subalgebras appearing in the quantum symmetric pairs of type AIII/AIV with no black nodes. As an application, we construct, for the 1st time, the canonical bases for the type B/C Schur algebras with unequal parameters associated with any weight function, using Lusztig’s bar-invariant basis [24] with unequal parameters.
Relation between Schur duality of type B/C at various specializations
| |$\mathbb{S}^{\bullet }_{n,d} \curvearrowright \mathbb{V}^{\otimes d}\curvearrowleft \mathbb{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[u^{\pm 1}, v^{\pm 1}] $| |
|---|---|
| |$\Downarrow \ \textrm{specialization at} \ u = \textbf{v}^{\textbf{L}(s_0)}, v = \textbf{v}^{\textbf{L}(s_1)}$| | |
| |$\mathbb{S}^{\bullet ,\textbf{L}}_{n,d} \curvearrowright \mathbb{V}_{\textbf{L}}^{\otimes d}\curvearrowleft \mathbb{H}^{\textbf{L}} _d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm \textbf{c}}]$| |
| |$\Downarrow \ \textrm{specialization at} \ u=v=\textbf{v} \ (\textrm{i.e.,} \ \textbf{L} = \ell )$| | |
| |$\textbf{S}^{\bullet }_{n,d} \curvearrowright \textbf{V}^{\otimes d}\curvearrowleft \textbf{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm 1}]$| |
| |$\mathbb{S}^{\bullet }_{n,d} \curvearrowright \mathbb{V}^{\otimes d}\curvearrowleft \mathbb{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[u^{\pm 1}, v^{\pm 1}] $| |
|---|---|
| |$\Downarrow \ \textrm{specialization at} \ u = \textbf{v}^{\textbf{L}(s_0)}, v = \textbf{v}^{\textbf{L}(s_1)}$| | |
| |$\mathbb{S}^{\bullet ,\textbf{L}}_{n,d} \curvearrowright \mathbb{V}_{\textbf{L}}^{\otimes d}\curvearrowleft \mathbb{H}^{\textbf{L}} _d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm \textbf{c}}]$| |
| |$\Downarrow \ \textrm{specialization at} \ u=v=\textbf{v} \ (\textrm{i.e.,} \ \textbf{L} = \ell )$| | |
| |$\textbf{S}^{\bullet }_{n,d} \curvearrowright \textbf{V}^{\otimes d}\curvearrowleft \textbf{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm 1}]$| |
Relation between Schur duality of type B/C at various specializations
| |$\mathbb{S}^{\bullet }_{n,d} \curvearrowright \mathbb{V}^{\otimes d}\curvearrowleft \mathbb{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[u^{\pm 1}, v^{\pm 1}] $| |
|---|---|
| |$\Downarrow \ \textrm{specialization at} \ u = \textbf{v}^{\textbf{L}(s_0)}, v = \textbf{v}^{\textbf{L}(s_1)}$| | |
| |$\mathbb{S}^{\bullet ,\textbf{L}}_{n,d} \curvearrowright \mathbb{V}_{\textbf{L}}^{\otimes d}\curvearrowleft \mathbb{H}^{\textbf{L}} _d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm \textbf{c}}]$| |
| |$\Downarrow \ \textrm{specialization at} \ u=v=\textbf{v} \ (\textrm{i.e.,} \ \textbf{L} = \ell )$| | |
| |$\textbf{S}^{\bullet }_{n,d} \curvearrowright \textbf{V}^{\otimes d}\curvearrowleft \textbf{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm 1}]$| |
| |$\mathbb{S}^{\bullet }_{n,d} \curvearrowright \mathbb{V}^{\otimes d}\curvearrowleft \mathbb{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[u^{\pm 1}, v^{\pm 1}] $| |
|---|---|
| |$\Downarrow \ \textrm{specialization at} \ u = \textbf{v}^{\textbf{L}(s_0)}, v = \textbf{v}^{\textbf{L}(s_1)}$| | |
| |$\mathbb{S}^{\bullet ,\textbf{L}}_{n,d} \curvearrowright \mathbb{V}_{\textbf{L}}^{\otimes d}\curvearrowleft \mathbb{H}^{\textbf{L}} _d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm \textbf{c}}]$| |
| |$\Downarrow \ \textrm{specialization at} \ u=v=\textbf{v} \ (\textrm{i.e.,} \ \textbf{L} = \ell )$| | |
| |$\textbf{S}^{\bullet }_{n,d} \curvearrowright \textbf{V}^{\otimes d}\curvearrowleft \textbf{H}_d$| | |$\textrm{over} \ {\mathbb{Z}}[\textbf{v}^{\pm 1}]$| |
The following diagram explains briefly the connection between the stabilization construction of type B/C for equal and unequal parameters (here |$\textbf{c} = \gcd (\textbf{L} (s_0), \textbf{L} (s_1))$|, and there are two distinct cases where |$\bullet $| can be replaced by |$\imath $| or |$\jmath $|).
At the specialization |$u=1$|, the Hecke algebra contains the type D Hecke algebra over |${\mathbb{Z}}(v^{\pm 1})$| as a proper subalgebra. Hence, the multiparameter Schur duality yields a weak Schur duality of type D that is used in [1] to formulate the Kazhdan–Lusztig theory for classical and super type D. The very duality also appears in [13] as a piece of a larger skew Howe duality of the quantum symmetric pair coideal subalgebra with itself.
1.3 Unequal parameters
While the organization of this paper follows closely to the (equal-parameter) affine-type C construction [16], the technical lemmas therein do not generalize naively. Below we mention some notable difficulties working with unequal parameters.
The 1st difficulty comes to dealing with the combinatorics of (type B/C) quantum numbers with two parameters. The key observation here is that the (equal-parameter) quantum numbers/factorials used in the BLM-type constructions arise from the (equal-parameter) Poincare polynomials corresponding to the Weyl groups. Hence, we compute the multiparameter upgrade for the type B/C Poincare polynomials (cf. Lemma 2.7), and then extract from it a type B/C quantum factorial (2.3.3) with two parameters.
The 2nd difficulty arises in constructing a standard basis of |${\mathbb{S}}^{\jmath }_{n,d}$|. For the equal-parameter case such a basis element |$[A]$| is obtained by multiplying a |$\textbf{v}$|-power to the evident basis |$e_A$|, while for unequal parameters, it is not obvious how to define a multiplier |$u^{\bullet } v^{\bullet }$| that specializes to the original |$\textbf{v}$|-power. We solve this problem by reducing it to getting an explicit formula (cf. Lemma 4.2) for the leading coefficient under the bar map. For the equal-parameter case the formula is obtained using certain identities on the dual Kazhdan–Lusztig basis due to Curtis. However, there are no multiparameter Kazhdan–Lusztig basis known to us (yet). Hence, we take a detour via Lusztig’s bar-invariant basis with unequal parameters and have successfully define a standard basis that affords the entire stabilization process.
Finally, we remark that there is an unexpected behavior for our multiparameter monomial bases—the basis elements are not bar-invariant, unlike the (equal-parameter) monomial basis elements. As a result, we can only show the existence of canonical bases for Schur algebras at certain specialization (see Section 4.4).
1.4 Organization and main results
We first start with the case |$\bullet = \jmath $|. In Section 2 we recall combinatorial properties of Weyl groups of type B/C in terms of permutation matrices. We characterize a matrix set |$\Xi _{n,d}$| (see (2.2.2)) associated with certain double coset representatives. We also introduce the multiparameter quantum numbers of type B/C corresponding to the Poincare polynomials. In Section 3 we introduce the Schur algebra |${\mathbb{S}}^{\jmath }_{n,d}$| (see (3.1.6)) with an evident basis |$\{e_A\mid A\in \Xi _{n,d}\}$|. In Section 4 we introduce a standard basis |$\{[A] \mid A\in \Xi _{n,d}\}$| (see (4.2.3)), and we show that, using Lusztig’s basis for the Hecke algebras with unequal parameters, it satisfies a unitriangular condition under the bar involution. The 1st main result is the following multiparameter upgrade of the multiplication formulas in [3]:
Let |$A, B \in \Xi _{n,d}$| and |$B - b(E_{h,h+1} +E_{-h,-h-1})$| is diagonal. Let |$\gamma _{B,A}^C \in{\mathbb{A}}$| be such that |$[B] [A] =\sum _C \gamma _{B,A}^C [C] \in{\mathbb{S}}^{\jmath }_{n,d}$|. The explicit formula and the vanishing criterion for |$\gamma _{B,A}^C$| are computed.
There exists a monomial basis |$\{m_A\}$| for the Schur algebra |${\mathbb{S}}^{\jmath }_{n,d}$| over |${\mathbb{A}}$|. Consequently, at a specialization associated with a weight function |$\textbf{L}$|, there exists a canonical basis |$\{\{A\}^{\textbf{L}} \}$| for |$\mathbb{S}^{\jmath , \textbf{L}}_{n,d}$|.
In Section 5 we show that the stabilization procedure along the line of Beilinson–Lusztig–MacPherson applies to the family of Schur algebras |$\{{\mathbb{S}}^{\jmath }_{n,d}\mid d \geqslant 1\}$| with a fixed |$n$|, which leads to the construction of stabilization algebra |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| (cf. Corollary 5.1.3) together with its canonical basis.
There exists a monomial basis |$\{m_A\}$| for the stabilization algebra |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|. As a corollary, there exists a canonical basis |$\{\{A\}^{\textbf{L}} \}$| for |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| at a specialization associated with a weight function |$\textbf{L}$|.
Section 6 is dedicated to the counterparts of Theorems B and C for the case |$\bullet = \imath $| (see Theorems 6.2.2 and 6.3.8). In Section 7 we show that the stabilization algebras coincide with the |$\mathfrak{g}\mathfrak{l}$|-variants |${\mathbb{U}}^{\jmath }, {\mathbb{U}}^{\imath }$| of the multiparameter quantum symmetric pair coideal subalgebras studied by Bao–Wang–Watanabe in [7] (referred as |$\textbf{U}^{\jmath }, \textbf{U}^{\imath }$| therein). The argument is made by passing the idempotented (or modified) quantum algebras.
There are |${\mathbb{Q}}(u,v)$|-algebra isomorphisms |$_{\mathbb{Q}}\dot{{\mathbb{K}}}^{\jmath }_{n} \simeq \dot{{\mathbb{U}}}^{\jmath }, {}_{\mathbb{Q}}\dot{{\mathbb{K}}}^{\imath }_{n} \simeq \dot{{\mathbb{U}}}^{\imath } $|, where |${}_{\mathbb{Q}} \dot{\mathbb{K}}_n^{\bullet }={\mathbb{Q}}(u,v)\otimes _{{\mathbb{A}}}\dot{\mathbb{K}}_n^{\bullet }$|.
In the appendix we provide an algebraic version of a type D Beilinson–Lusztig–MacPherson construction that is first introduced by Fan–Li from a geometric viewpoint.
2 Combinatorics on Weyl Groups
2.1 Weyl groups as permutation groups
Let |$\ell :W\to{\mathbb{N}}$| be the length function on |$W$|. We introduce a truncated length function |$\ell _{\mathfrak{c}}:W \to{\mathbb{N}}$| such that |$\ell _{\mathfrak{c}}(g)$| equals to the total number of |$s_0$|’s in a reduced expression of |$g$|. The function |$\ell _{\mathfrak{c}}$| is well defined since it is the weight function (cf. [24]) determined by |$\ell _{\mathfrak{c}}(s_0) = 1, \ell _{\mathfrak{c}}(s_i) = 0$| for |$i\geqslant 1$|. We set |$\ell _{\mathfrak{a}} = \ell - \ell _{\mathfrak{c}}$|.
It follows by an easy induction that |$\ell _{\mathfrak{c}}(g) = {}^{\sharp }\{ i\in [1, d] | g(i) < 0\}$|, which yields to (2.1.5) by a direct calculation. The formula (2.1.7) for |$\ell (g)$| is equivalent to the formula [2, (8.2)]. Then there comes the formula (2.1.6) by |$\ell _{\mathfrak{a}}(g)=\ell (g)-\ell _{\mathfrak{c}}(g)$|.
The expressions in Lemma 2.1.1 are not the most straightforward. There are simpler ones, for example, |$\ell _{\mathfrak{a}} = \textrm{inv} +\textrm{neg}$| and |$\ell _{\mathfrak{c}} = \textrm{neg}$| following the convention in [2]. We will see in Lemma 2.2.2 the advantage of choosing such symmetrized expressions. See also [16, Appendix A] for similar symmetrized length formulas for finite and affine classical types.
In the following we collect some standard results for Coxeter groups from [10, Proposition 4.16, Lemma 4.17, and Theorem 4.18].
Let |$\lambda ,\mu \in \Lambda _{n,d}$| and |$g \in \mathscr{D}_{\lambda \mu }$|.
(a) There exists |$\delta \in \Lambda _{n^{\prime },d}$| for some |$n^{\prime }$| such that |$W_{\delta } = g^{-1} W_\lambda g \cap W_\mu .$|
(b) The map |$W_\lambda \times (\mathscr{D}_\delta \cap W_\mu ) \rightarrow W_\lambda g W_\mu $| sending |$(x,y)$| to |$xgy$| is a bijection; moreover, we have |$\ell (xgy) = \ell (x) + \ell (g) + \ell (y)$|.
(c) The map |$W_\delta \times (\mathscr{D}_\delta \cap W_\mu ) \rightarrow W_\mu $| sending |$(x,y)$| to |$xy$| is a bijection; moreover, we have |$\ell (x) + \ell (y) = \ell (xy)$|.
An essential step in deriving the multiplication formula is to understand the set |$\mathscr{D}_\delta \cap W_\mu $|, which we will see in Section 3.2.
2.2 Set-valued matrices
For |$A=(a_{ij}) \in \Xi _{n,d}$| we define a matrix |$A^{\mathcal{P}} = (A^{\mathcal{P}}_{ij})$| to be the unique set-valued matrix satisfying the following:
(P0) The sets |$(A^{\mathcal{P}}_{ij})_{ij}$| partition |$[-d,d]$|;
(P1) |$|A^{\mathcal{P}}_{ij}| = a_{ij}$| for all |$i,j$|;
(P2) Every element in |$A^{\mathcal{P}}_{ij}$| is smaller than any element in |$A^{\mathcal{P}}_{xy}$| if |$(i,j) < (x,y)$| in the lexicographical order (i.e., |$(i,j) < (x,y)$| if and only if |$i<x$| or |$(i=x, j < y)$|).
The surjectivity follows from |$\kappa (\textrm{row}(A), g_A, \textrm{col}(A))=A\ (\forall A\in \Xi _{n,d})$| by a direct calculation.
For injectivity, we assume |$\kappa (\lambda ,g,\mu ) = A = \kappa (\lambda ^{\prime },g^{\prime },\mu ^{\prime })$|. Then |$\lambda =\lambda ^{\prime } = \textrm{row}(A)$| and |$\mu = \mu ^{\prime } = \textrm{col}(A)$| and hence |$g,g^{\prime }\in \mathscr{D}_{\lambda \mu }$|. It follows from |$|R^{\lambda }_i \cap g R^{\mu }_j|=|R^{\lambda }_i \cap g^{\prime } R^{\mu }_j|\ (\forall i,j\in [-n,n])$| that |$g = w_{(\lambda )} g^{\prime } w_{(\mu )}$| for some |$w_{(\lambda )} \in W_{\lambda }, w_{(\mu )} \in W_{\mu }$|. Therefore, |$g = g^{\prime }$| since they are both minimal double coset representatives in |$W_\lambda \backslash W / W_\mu $|.
These three formulas are paraphrases of those in Lemma 2.1.1.
A direct computation shows that |$\delta (A)$| is indeed a weak composition |$\delta $| in Lemma 2.1.3(a).
2.3 Quantum combinatorics
Let |$A = \kappa (\mu ,g,\nu )$|, and let |$\delta = \delta (A)$|. Then |$\sum \limits _{w \in W_{\delta }} u^{2\ell _{\mathfrak{c}}(w)}v^{2\ell _{\mathfrak{a}}(w)} = [A]_{\mathfrak{c}}^{!}$|.
Let |$W^{\mathfrak{c}}_d$| be the Weyl group of type C|$_d$|.
3 Schur Algebras
3.1 Schur algebras
If |$w \in W_\lambda $|, then |$T_w x_\lambda = u^{2\ell _{\mathfrak{c}}(w)}v^{2\ell _{\mathfrak{a}}(w)} x_\lambda = x_\lambda T_w$|.
This reduces to the case when |$w = s \in S$|. It then follows from the Hecke relation (3.1.1).
The set |$\{ e_A | A\in \Xi _{n,d} \}$| forms an |${\mathbb{A}}$|-basis of |${\mathbb{S}}^{\jmath }_{n,d}$|.
|$x_\mu T_{g} x_\nu = [A]^!_{\mathfrak{c}} \, e_A(x_\nu ).$|
3.2 Multiplication formulas |$\mathscr{D}_\delta \cap W_\mu $|
The lemma follows from (3.2.3).
Suppose that |$A, B, C\in \Xi _{n,d}$| and |$h\in [1,n]$|.
- (1) If |$B-bE_{h,h-1}^{\theta }$| is diagonal, |$\textrm{col}(B)=\textrm{row}(A)$|, thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=b$| such that(3.2.6)$$\begin{equation} e_B e_A=\sum_{t}v^{2\sum_{k<l}t_la_{h,k}}\prod_{l=-n}^{n} \left[\begin{array}{cc} a_{h,l}+t_l\\ t_l \end{array}\right] e_{{A}_{t,h}}, \end{equation}$$and(3.2.7)$$\begin{equation} \begin{cases} t_i\leqslant a_{h-1,i} & \textrm{if} \ h>1;\\ t_i+t_{-i}\leqslant a_{h-1,i} &\textrm{if} \ h=1, \end{cases} \end{equation}$$$$\begin{equation*} {A}_{t,h}=A+\sum_{l=-n}^{n}t_lE_{h,l}^{\theta}-\sum_{l=-n}^{n}t_lE_{h-1,l}^{\theta}. \end{equation*}$$
- (2) Suppose |$C-ce:{h-1,h}^{\theta }$| is diagonal and |$\textrm{col}(C)=\textrm{row}(A)$|. If |$h\neq 1$|, thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=c$| such that |$t_i\leqslant a_{h,i}$|, and(3.2.8)$$\begin{equation} e_C e_A=\sum_{t}v^{2\sum_{k>l}t_la_{h-1,k}}\prod_{l=-n}^{n}\left[\begin{array}{cc}a_{h-1,l}+t_l\\t_l\end{array}\right] e_{\widehat{A}_{t,h}}, \end{equation}$$If |$h=1$|, then$$\begin{equation*} \widehat{A}_{t,h}=A-\sum_{l=-n}^{n}t_lE_{h,l}^{\theta}+\sum_{l=-n}^{n}t_lE_{h-1,l}^{\theta}. \end{equation*}$$where |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=c$| such that |$t_i\leqslant a_{1,i}$|.(3.2.9)$$\begin{align} e_C e_A=& \sum_{t} u^{2\sum_{l<0}t_l} v^{2\sum_{k>l} a_{0,k}t_l+2\sum_{l<k<-l}t_lt_k+\sum_{l<0}t_l(t_l-3)}\nonumber\\& \frac{[a_{0,0}^{\natural}+t_0]_{\mathfrak{c}}^!}{[a_{0,0}^{\natural}]_{\mathfrak{c}}^![t_0]!}\prod_{l=1}^n\frac{[a_{0,l}+t_l+t_{-l}]!}{[a_{0,l}]![t_l]![t_{-l}]!}e_{\widehat{A}_{t,1}}, \end{align}$$
These explicit formulas match the ones in [3] (resp. the unsigned ones in [14]) if we specialize |$u=v$| (resp. |$u=1$|).
4 Canonical Bases
4.1 The bar involution
For |$\lambda ,\mu \in \Lambda _{n,d}$| (see (2.1.8)), let |$g^+_{\lambda \mu }$| be the longest element in the double coset |$W_\lambda g W_\mu $| for |$g \in \mathscr{D}_{\lambda \mu }$|, and let |$w_\circ ^\mu = \mathbb{1}_{\mu \mu }^+$| be the longest element in the parabolic subgroup |$W_\mu = W_\mu \mathbb{1} W_\mu $|. The lemma below is standard (cf. [10, Corollary 4.19]).
Let |$A = \kappa (\lambda ,g,\mu )$|, |$\delta = \delta (A)$|. Then
(a) |$g_{\lambda \mu }^+ = w_\circ ^{\lambda } g w_\circ ^{\delta } w_\circ ^{\mu }$|, and |$\ell (g_{\lambda \mu }^+) = \ell (w_\circ ^{\lambda }) + \ell (g) - \ell (w_\circ ^{\delta }) + \ell (w_\circ ^{\mu }).$|
(b) |$W_\lambda g W_\mu = \{w \in W | g \leqslant w \leqslant g^+_{\lambda \mu }\}$|.
(C1) |$C^{\prime }_w$| is bar-invariant;
(C2) |$C^{\prime }_w = v^{-\ell (w)}\sum _{y \leqslant w} P_{yw}(v) T_y$|.
4.2 A standard basis in |${\mathbb{S}}^{\jmath }_{n,d}$|
The function |$\widehat{\ell }$| counts the dimension of the generalized Schubert variety associated with the matrix |$A$| (cf. [16, Appendix A]) and is equal to the length of |$A$| when |$A$| is a permutation matrix (that is when the associated variety is a genuine Schubert variety).
Let |$A =\kappa (\lambda ,g,\mu )\in \Xi _{n,d}$|. Then we have |$\overline{[A]} \in [A] +\sum _{B <_{\textrm{alg}} A} {\mathbb{A}} [B].$|
Let us reformulate the multiplication formula for |${\mathbb{S}}^{\jmath }_{n,d}$| (Proposition 3.2.2) in terms of the standard basis.
Suppose that |$A, B, C\in \Xi _{n,d}$|, and |$h\in [1,n]$|.
- (1) If |$B-bE_{h,h-1}^{\theta }$| is diagonal, |$\textrm{col}(B)=\textrm{row}(A)$|, thenwhere |$t$| is summed over as in Proposition 3.2.2 (1), and(4.2.10)$$\begin{equation} [B] [A]=\sum_{t}u^{-\delta_{h,1}\sum_{l>0}{t_l}}v^{\beta(t)}\prod_{l=-n}^{n}\overline{\left[\begin{array}{cc}a_{h,l}+t_l\\t_l\end{array}\right]} [\widehat{A}_{t,h}], \end{equation}$$(4.2.11)$$\begin{equation} \beta(t)=\sum_{k\leqslant l}t_la_{h,k}-\sum_{k<l}t_l(a_{h-1,k}-t_k)+\delta_{h,1}(\sum_{-l<k<l}t_lt_k+\sum_{l>0}\frac{t_l(t_l+3)}{2}). \end{equation}$$
- (2) Suppose |$C-ce:{h-1,h}^{\theta }$| is diagonal and |$\textrm{col}(C)=\textrm{row}(A)$|. If |$h\neq 1$| thenwhere |$t$| is summed over as in Proposition 3.2.2 (2), and(4.2.12)$$\begin{equation} [C] [A]=\sum_{t}v^{\beta^{\prime}(t)}\prod_{l=-n}^{n}\overline{\left[\begin{array}{cc}a_{h-1,l}+t_l\\t_l\end{array}\right]} [\widehat{A}_{t,h}], \end{equation}$$If |$h=1$| then(4.2.13)$$\begin{equation} \beta^{\prime}(t)=\sum_{k\geqslant l}t_la_{h-1,k}-\sum_{k>l}t_l(a_{h,k}-t_k). \end{equation}$$where(4.2.14)$$\begin{equation} [C][A]=\sum_{t}u^{\sum_{l\leqslant0}t_l}v^{\beta^{\prime\prime}(t)} \overline{\left(\frac{[a_{0,0}^{\natural}+t_0]_{\mathfrak{c}}^!}{[a_{0,0}^{\natural}]_{\mathfrak{c}}^![t_0]!}\prod_{l=1}^n \frac{[a_{0,l}+t_l+t_{-l}]!}{[a_{0,l}]![t_l]![t_{-l}]!}\right)}[\widehat{A}_{t,1}], \end{equation}$$(4.2.15)$$\begin{equation} \beta^{\prime\prime}(t)=\sum_{k\geqslant l}t_la_{0,k}-\sum_{k>l}t_l(a_{1,k}-t_k)+\sum_{l<k\leqslant -l}t_lt_k+\sum_{l\leqslant 0}\frac{t_l(t_l-3)}{2}. \end{equation}$$
4.3 A monomial basis in |${\mathbb{S}}^{\jmath }_{n,d}$|
The monomial basis acts as an intermediate step toward constructing canonical basis in the one-parameter case. Moreover, the two-parameter stabilization procedure is made possible thanks to the property (4.3.3) of monomial basis.
4.4 The canonical basis at the specialization
Let |$\mathbb{S}^{\jmath , \textbf{L}} _{n,d}$| be the specialization of |${\mathbb{S}}^{\jmath }_{n,d}$| at |$u=\textbf{v}^{\textbf{L} (s_0)}, v=\textbf{v}^{\textbf{L} (s_1)}$|.
There exists a canonical basis |$\{\{A\}^{\textbf{L}} \ |\ A \in \Xi _{n,d}\}$| for |$\mathbb{S}^{\jmath , \textbf{L}} _{n,d}$|, which is characterized by the property (4.4.3).
5 Stabilization Algebra |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|
In this section, we shall establish a stabilization property for the family of Schur algebras |${\mathbb{S}}^{\jmath }_{n,d}$| as |$d$| varies, which leads to a quantum algebra |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|.
5.1 A BLM-type stabilization
By an argument identical with [4, Proposition 4.3], we obtain below the stabilization of bar involution by allowing extra coefficients as seen in (5.1.6).
Let |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| be the free |${\mathbb{A}}$|-module with an |${\mathbb{A}}$|-basis given by the symbols |$[A]$| for |$A\in \widetilde{\Xi }_n$| (which will be called a standard basis of |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|). By Propositions 5.1 and -5.2 and applying a specialization at |$\pi =1$| (note that |$\zeta _i(u,v, 1), \tau _i(u,v, 1)\in{\mathbb{A}}$| because of |$r^{(1)}_{a,k}(u,v,1), r^{(2)}_{a,k}(u,v,1)\in{\mathbb{A}}$|), we have the following corollary.
The following multiplication formula in |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| follows directly from Theorem 4.5 by the stabilization construction.
Let |$A, B, C \in \widetilde{\Xi }_{n}$|, and |$h\in [1,n]$|.
- (1) If |$B-bE_{h,h-1}^{\theta }$| is diagonal and |$\textrm{col}(B)=\textrm{row}(A)$|, thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=b$| such that(5.1.9)$$\begin{equation} [B] [A]=\sum_{t}u^{-\delta_{h,1}\sum_{l>0}{t_l}}v^{\beta(t)}\prod_{l=-n}^{n}\overline{\left[\begin{array}{cc}a_{h,l}+t_l\\t_l\end{array}\right]} [ {A}_{t,h}], \end{equation}$$$$\begin{equation*}\left\{\begin{array}{ll} t_i\leqslant a_{h-1,i} & \ \textrm{if } i+1\neq h>1;\\ t_i+t_{-i}\leqslant a_{0,i} & \ \textrm{if } h=1, i\neq0; \end{array} \right.\end{equation*}$$
(2) Suppose |$C-ce:{h-1,h}^{\theta }$| is diagonal and |$\textrm{col}(C)=\textrm{row}(A)$|.
If |$h\neq 1$| thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=c$| such that |$t_i\leqslant a_{h,i}$| if |$i\neq h$|.(5.1.10)$$\begin{equation} [C] [A]=\sum_{t}v^{\beta^{\prime}(t)}\prod_{l=-n}^{n}\overline{\left[\begin{array}{cc}a_{h-1,l}+t_l\\t_l\end{array}\right]} [\widehat{A}_{t,h}], \end{equation}$$If |$h=1$| thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=c$| such that |$t_i\leqslant a_{1,i}$| if |$i\neq 1$|.(5.1.11)$$\begin{equation} [C][A]=\sum_{t}u^{\sum_{l\leqslant0}t_l}v^{\beta^{\prime\prime}(t)} \overline{\left(\frac{\prod_{k=a^{\natural}_{00}+1}^{a^{\natural}_{00}+t_{0}}[k](u^2v^{2(k-1)}+1)} {\prod_{k=1}^{t_{0}}[k]}\prod_{l=1}^n\frac{[a_{0,l}+t_l+t_{-l}]!}{[a_{0,l}]![t_l]![t_{-l}]!}\right)}[\widehat{A}_{t,1}], \end{equation}$$
5.2 Monomial and canonical bases for |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|
The proposition below follows from Proposition 4.3.1 by the stabilization construction.
There exists a canonical basis |$\dot{{\mathfrak{B}}} = \{\{A\}^{\textbf{L}} \ |\ A \in \widetilde{\Xi}_{n}\}$| for |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| at the specialization |$u=\textbf{v}^{\textbf{L} (s_0)}, v=\textbf{v}^{\textbf{L} (s_1)}$|, which is characterized by the property (4.4.3).
6 A Different Stabilization Algebra |$\dot{{\mathbb{K}}}^{\imath }_{n}$|
In this section we formulate a variant of Schur algebras and their corresponding stabilization algebras. We construct the distinguished bases of these algebras. Recall |$N=2n+1$|.
6.1 |$\imath $|-Schur algebras
The map |$\kappa ^{\imath }: \bigsqcup _{\lambda ,\mu \in \Lambda _{n,d}^{\imath }} \{\lambda \} \times \mathscr{D}_{\lambda \mu } \times \{\mu \} \longrightarrow \Xi ^{\imath }$| sending |$(\lambda , g, \mu )$| to |$(|R_i^{\lambda } \cap g R_j^\mu |)$| is a bijection.
6.2 Monomial and canonical bases for |${\mathbb{S}}^{\imath }_{n,d}$|
For each |$A \in \Xi ^{\imath }$|, we have |$m_A\in{\mathbb{S}}^{\imath }_{n,d}$|. Hence, the set |$\{m_A | A \in \Xi ^{\imath } \}$| forms an |${\mathbb{A}}$|-basis of |${\mathbb{S}}^{\imath }_{n,d}$|. Furthermore, we have |$m_A \in [A] +\sum _{B\in \Xi ^{\imath }, B<_{\textrm{alg}} A} {\mathbb{A}} [B]$|.
At the specialization |$u = \textbf{v}^{\textbf{L}} (s_0), v= \textbf{v}^{\textbf{L}} (s_1)$|, there is a canonical basis |$\mathfrak B_{n,d}^{\imath } = \{ \{A\}^{\textbf{L}} | A \in \Xi ^{\imath } \}$| of |${\mathbb{S}}^{\imath }_{n,d}$| such that |$\overline{\{A\}^{\textbf{L}}} \ =\{A\}^{\textbf{L}} $| and |$\{A\}^{\textbf{L}} \in [A]^{\textbf{L}} + \sum _{B\in \Xi ^{\imath }, B<_{\textrm{alg}} A} \textbf{v}^{-\textbf{c}} {\mathbb{Z}}[\textbf{v}^{-\textbf{c}}] [B]^{\textbf{L}} $|. Moreover, we have |$\mathfrak B_{n,d}^{\imath } = \mathfrak B_{n,d}^{\jmath }\cap{\mathbb{S}}^{\imath }_{n,d}$|.
The 1st half statement on the canonical basis follows by Proposition 6.2 and a standard argument (cf. [22, 24.2.1]). The 2nd half statement follows from the uniqueness characterization of the canonical basis |$\mathfrak B_{n,d}^{\imath }$|.
6.3 Stabilization algebra of type |$\imath $|
The proof is similar to the proof of Proposition 5.1 where |${_{p}}{A} = A+pI$| is used instead of |${_{\breve{p}}}{A}$|.
Precisely, we have the following multiplication formulas for Chevalley generators in |$\dot{{\mathbb{K}}}^{>}_n $|.
Let |$A, B, C \in \widetilde{\Xi }_n^>$|, and |$h\in [1,n]$|.
- (1) If |$B-bE_{h,h-1}^{\theta }$| is diagonal and |$\textrm{col}(B)=\textrm{row}(A)$|, thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=b$| such that(6.3.4)$$\begin{equation} [B] [A]=\sum_{t}u^{-\delta_{h,1}\sum_{l>0}{t_l}}v^{\beta(t)}\prod_{l=-n}^{n}\overline{\left[\begin{array}{cc}a_{h,l}+t_l\\t_l\end{array}\right]} [ {A}_{t,h}], \end{equation}$$$$\begin{equation*} \left\{\begin{array}{ll} t_i\leqslant a_{h-1,i} & \ \textrm{if } i+1\neq h>1;\\ t_i+t_{-i}\leqslant a_{0,i} & \ \textrm{if } h=1, \forall i; \end{array} \right. \end{equation*}$$
- (2) Suppose |$C-ce:{h-1,h}^{\theta }$| is diagonal and |$\textrm{col}(C)=\textrm{row}(A)$|.If |$h\neq 1$| thenwhere |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=c$| such that |$t_i\leqslant a_{h,i}$| if |$i\neq h$|.If |$h=1$| then(6.3.5)$$\begin{equation} [C] [A]=\sum_{t}v^{\beta^{\prime}(t)}\prod_{l=-n}^{n}\overline{\left[\begin{array}{cc}a_{h-1,l}+t_l\\t_l\end{array}\right]} [\widehat{A}_{t,h}], \end{equation}$$where |$t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N$| with |$\sum _{i=-n}^n t_i=c$| such that |$t_i\leqslant a_{1,i}$| if |$i\neq 1$|.(6.3.6)$$\begin{equation} [C][A]=\sum_{t}u^{\sum_{l\leqslant0}t_l}v^{\beta^{\prime\prime}(t)} \overline{\left(\frac{\prod_{k=a^{\natural}_{00}+1}^{a^{\natural}_{00}+t_{0}}[k](u^2v^{2(k-1)}+1)} {\prod_{k=1}^{t_{0}}[k]}\prod_{l=1}^n\frac{[a_{0,l}+t_l+t_{-l}]!}{[a_{0,l}]![t_l]![t_{-l}]!}\right)}[\widehat{A}_{t,1}], \end{equation}$$
The set |$\dot{{\mathbb{K}}}^{\imath }_{n} \cap \dot{\mathfrak B}^{\jmath ,>}$| forms a canonical basis of |$\dot{{\mathbb{K}}}^{\imath }_{n}$|.
The submodule of |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| spanned by |$[A]$| for |$A \in \widetilde{\Xi ^{\imath }}$| is not a subalgebra. This is why we need a somewhat different stabilization above to construct the canonical basis for |$\dot{{\mathbb{K}}}^{\imath }_{n}$|. We shall see below the stabilization above is related to the stabilization used earlier. Define |${\mathbb{J}}$| to be the |${\mathbb{A}}$|-submodule of |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| spanned by |$[A]$| for all |$A \in \widetilde{\Xi }_n^<$|.
The submodule |${\mathbb{J}}$| is a two-sided ideal of |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|.
If |$A \in \widetilde{\Xi }_n^<$| then |$m_A \in{\mathbb{J}}$|.
The proof is as the same as the one of [3, Lemma A.6 (1)].
Recall |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| admits a canonical basis of |$\dot{{\mathfrak{B}}}$| at the specialization |$u=\textbf{v}^{\textbf{L} (s_0)}, v=\textbf{v}^{\textbf{L} (s_1)}$| from Theorem 5.2.2.
The ideal |${\mathbb{J}}$| admits a monomial basis |$\{m_A | A \in \widetilde{\Xi }_n^<\}$|. Moreover, its specialization at |$u=\textbf{v}^{\textbf{L} (s_0)}, v=\textbf{v}^{\textbf{L} (s_1)}$| (denoted by |${\mathbb{J}}^{\textbf{L}} $|) has a canonical basis |$\dot{{\mathfrak{B}}} \cap{\mathbb{J}}^{\textbf{L}} = \{\{A\}^{\textbf{L}} | A \in \widetilde{\Xi }_n^<\}$|.
The 1st statement follows from the above lemma directly. Since |$m_A=[A]+ \ \textrm{lower terms}$|, we know that |${\mathbb{J}}^{\textbf{L}} $| is bar invariant. Thus, |${\mathbb{J}}^{\textbf{L}} $| does admit a canonical bases parameterized by |$A \in \widetilde{\Xi }_n^<$|, which should be |$\dot{{\mathfrak{B}}} \cap{\mathbb{J}}^{\textbf{L}} = \{\{A\}^{\textbf{L}} | A \in \widetilde{\Xi }_n^<\}$| by the uniqueness of canonical basis.
The following statements hold:
(a) The quotient algebra |$\dot{{\mathbb{K}}}^{\jmath }_{n}/{\mathbb{J}}$| admits a monomial basis |$\{m_A + {\mathbb{J}} | A\in \widetilde{\Xi }_n^>\}$|.
(b) The specialization at |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$| of the quotient algebra |$\dot{{\mathbb{K}}}^{\jmath }_{n}/{\mathbb{J}}$| admits a canonical basis |$\{\{A\}^{\textbf{L}} + {\mathbb{J}}^{\textbf{L}} | A \in \widetilde{\Xi }_n^>\}$|.
(c) The map |$\sharp : \dot{{\mathbb{K}}}^{\jmath }_{n}/{\mathbb{J}} \rightarrow \dot{{\mathbb{K}}}^{>}_n $| sending |$[A] + {\mathbb{J}} \mapsto [A]$| is an isomorphism of |${\mathbb{A}}$|-algebras, which matches the corresponding monomial bases. It also matches the corresponding canonical bases at the specialization |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$|.
Parts (a) and (b) follow directly from Theorem 6.3.6. Below we prove the Part (c). Knowing that the map |$\sharp $| is a linear isomorphism, we need to verify it is an algebraic homomorphism. Comparing the multiplication formulas for |$\dot{{\mathbb{K}}}^{\jmath }_{n}$| in Proposition 5.1.4 with the ones for |$\dot{{\mathbb{K}}}^{>}_n $| in Proposition 6.3.2, we can see that the structure constants with respect to the Chevalley generators for |$\dot{{\mathbb{K}}}^{\jmath }_{n}/{\mathbb{J}}$| are as the same as those for |$\dot{{\mathbb{K}}}^{>}_n $|. Therefore, |$\sharp $| is an algebraic homomorphism.
Since |$\sharp $| matches the Chevalley generators, it matches the corresponding monomial bases. We also obtain that |$\sharp $| commutes with the bar involution. Notice that the partial orders |$<_{\textrm{alg}}$| are compatible; hence, |$\sharp $| also matches the corresponding canonical bases at the specialization |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$|.
We summarize Lemma 6.3.3 and Proposition 6.3.7 above as follows.
As an |${\mathbb{A}}$|-algebra, |$\dot{{\mathbb{K}}}^{\imath }_{n}$| is isomorphic to a subquotient of |$\dot{{\mathbb{K}}}^{\jmath }_{n}$|, with compatible standard, monomial basis. They have compatible canonical bases at the specialization |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$|.
(a) The monomial basis of |${\dot{{\mathbb{K}}}^{\jmath }_{n}}$| restricts to the monomial basis of |$\dot{{\mathbb{K}}}_n^{\jmath ,1}$|; the monomial basis of |$\dot{{\mathbb{K}}}_n^{\jmath ,1}$| restricts to the monomial basis of |${\mathbb{J}}^1$|. So does the canonical basis at the specialization |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$|.
(b) The quotient |${\mathbb{A}}$|-subalgebra |$\dot{{\mathbb{K}}}_n^{\jmath ,1}/{\mathbb{J}}^1$| admits a monomial basis |$\{m_A + {\mathbb{J}}^1 | A\in \widetilde{\Xi ^{\imath }}\}$|. It also admits a canonical basis |$\{\{A\}^{\textbf{L}} + {\mathbb{J}}^{1,\textbf{L}} \ | A \in \widetilde{\Xi ^{\imath }}\}$| at the specialization |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$|, where |${\mathbb{J}}^{1,\textbf{L}} \ ={\mathbb{J}}^1|_{u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}}$|.
(c) There is an |${\mathbb{A}}$|-algebra isomorphism |$\dot{{\mathbb{K}}}_n^{\jmath ,1}/{\mathbb{J}}^1\cong \dot{{\mathbb{K}}}^{\imath }_{n}$|, which matches the corresponding monomial bases. It also matches the corresponding canonical basis at the specialization |$u = \textbf{v}^{\textbf{L} (s_0)}, v= \textbf{v}^{\textbf{L} (s_1)}$|.
7 Quantum Symmetric Pairs
7.1 The quantum symmetric pair |$({\mathbb{U}},{\mathbb{U}}^{\jmath })$|
We start with the quantum symmetric pairs of type AIII/AIV without fixed points nor black nodes, associated with the following Satake diagram:
The (multiparameter) quantum symmetric pairs |$({\mathbb{U}},{\mathbb{U}}^{\jmath })$| in this paper are the |$\mathfrak{g}\mathfrak{l}$|-variant of the quantum symmetric pairs in [7].
7.2 Isomorphism |$\dot{\mathbb{U}}^{\jmath }\simeq \dot{{\mathbb{K}}}^{\jmath }_{n}$|
Set |${}_{\mathbb{Q}}\dot{{\mathbb{K}}}^{\jmath }_{n}={\mathbb{Q}}(u,v)\otimes _{\mathbb{A}} \dot{{\mathbb{K}}}^{\jmath }_{n}$|.
We also know that |$\aleph $| is a linear isomorphism. The argument is almost as the same as that for the case of specialization at |$u=v$|, which can be found in the proof of [3, Theorem 4.7]. Therefore, |$\aleph $| is an isomorphism of |${\mathbb{Q}}(u,v)$|-algebras.
7.3 The quantum symmetric pair |$({\mathbb{U}},{\mathbb{U}}^{\imath })$|
Below we formulate the counterparts of Sections 7.1 and 7.2. The proofs are very similar and will often be omitted. We now work on quantum symmetric pairs of type AIII with fixed points associated with the Satake diagram below:
It was observed in [7, 20] that the parameter |$\omega \in{\mathbb{Q}}(u,v)$| in the embedding |$t=E_0+vF_0K_0^{-1}+\omega K_0^{-1}$| is irrelevant to the presentation of the algebra |${\mathbb{U}}^{\imath }$|.
An argument similar to the proof of [3, Theorem A.15] also shows |$\aleph $| is a linear isomorphism. Therefore, |$\aleph $| is an isomorphism of |${\mathbb{Q}}(u,v)$|-algebras.
Appendix A. An Algebraic Approach to Schur Algebras of Type D
As we mentioned in Section 2, at the specialization |$u=1$| the multiparameter Schur duality yields a weak Schur duality of type D that is used in [1] to formulate the Kazhdan–Lusztig theory for classical and super type D. These algebras |$\mathbb{S}^{\bullet }_{n,d}|_{u=1}$| (|$\bullet = \imath $| or |$\jmath $|), however, are not the Schur algebras introduced in [14]. While bases of Schur algebras of finite type A/B/C and affine type A/C can be parametrized by a matrix set (cf. |$\Xi _{n,d}$| in 2.2.2), for finite type D Fan and Li showed that a matrix set is not enough—a notion of signed matrices that indexes a larger algebra is needed. From a geometric point of view, this reflects the fact that there are two connected components for the maximal isotropic Grassmannian associated with |$\textrm{SO}(2d)$|. In this appendix, we provide an algebraic approach to Fan–Li’s construction parallel to our multiparameter results. With our algebraic approach we also clear up some previous misconceptions (e.g., compare (A.4.2) and [14, (22)]). The arguments are very similar to the multiparameter counterpart, so we will omit the easy proofs in this appendix.
A.1 Weyl groups of type |$\textbf{D}$|
A.2 Signed compositions
Let |$g \in W_{\textbf{D}}$| and |$\lambda ^{\alpha } \in \Lambda _{\textbf{D}}$|.
(a) If |$\alpha =\pm $|, then |$g\in \mathscr{D}_{\lambda ^{\alpha }}$| if and only if |$g^{-1}$| is order-preserving on |$R^{\lambda ^{\alpha }}_i$|, for all |$i \in [1,n]$|;
- (b) If |$\alpha =0$|, then |$g\in \mathscr{D}_{\lambda ^{\alpha }}$| if and only if |$g^{-1}$| is order-preserving on |$R^{\lambda ^{\alpha }}_i$| for all |$i \in [1,n]$| and$$\begin{equation*} g^{-1}(-2)<g^{-1}(1)<g^{-1}(2)<\cdots<g^{-1}(\lambda_0). \end{equation*}$$
By a similar argument for [10, Proposition 4.16, Lemma 4.17, and Theorem 4.18], we have the following facts.
Let |$\lambda ^{\alpha },\mu ^{\beta } \in \Lambda _{\textbf{D}}$|, and |$g \in \mathscr{D}_{\lambda ^{\alpha }\mu ^{\beta }}$|.
(a) There is a weak composition |$\delta = \delta (\lambda ^{\alpha }, g, \mu ^{\beta }) \in \Lambda _{n^{\prime},d}$| for some |$n^{\prime}$| such that |$W_{\delta ^{\beta }} = g^{-1} W_{\lambda ^{\alpha }} g \cap W_{\mu ^{\beta }}$|.
(b) The map |$W_{\lambda ^{\alpha }} \times (\mathscr{D}_\delta \cap W_{\mu ^{\beta }}) \rightarrow W_{\lambda ^{\alpha }} g W_{\mu ^{\beta }}$| sending |$(x,y)$| to |$xgy$| is a bijection; moreover, we have |$\ell (xgy) = \ell (x) + \ell (g) + \ell (y)$|.
(c) The map |$(\mathscr{D}_\delta \cap W_{\mu ^{\beta }}) \times W_\delta \rightarrow W_{\mu ^{\beta }}$| sending |$(x,y)$| to |$xy$| is a bijection; moreover, we have |$\ell (x) + \ell (y) = \ell (xy)$|.
A.3 Schur algebras
The set |$\{\phi _{\lambda ^{\alpha }\mu ^{\beta }}^g | (\lambda ^{\alpha },g, \mu ^{\beta }) \in \mathscr{D}_{n,d} \}$| forms an |$\textbf{A}$|-basis of |$\textbf{S} _{n,d}$|.
A.4 Signed matrices
The map |$\kappa :\mathscr{D}_{n,d}\rightarrow \Xi _{\textbf{D}}$| is a bijection.
Let |${\mathcal{A}}=\kappa (\lambda ^{\alpha },g,\mu ^{\beta })\in \Xi _{\textbf{D}}$|, then |$p({\mathcal{A}})=+$| (resp. −) if and only if |$g(1)>0$| (resp. |$<0$|).
A direct computation shows that |$\delta ({\mathcal{A}})$| is indeed a weak composition |$\delta $| in Proposition A.2.3(a).
Let |${\mathcal{A}}=\kappa (\lambda ^{\alpha },g,\mu ^{\beta })\in \Xi _{\textbf{D}}$|. Then |$W_{\delta ({\mathcal{A}})} = g^{-1} W_{\lambda ^{\alpha }} g \cap W_{\mu ^{\beta }}.$|
For any |${\mathcal{A}}=A^{\alpha }\in \Xi _{\textbf{D}}$| with |$A=(a_{ij})$|, we have |$\sum _{w\in W_{\delta ({\mathcal{A}})}}v^{2\ell (w)}=[A]_{\mathfrak{d}}^{!}.$|
A.5 Multiplication formulas
The proofs of Lemma A.5.1–A.5.3 are very similar to their counterparts (Lemma 3.1.3, (3.2.2) and Lemma 3.2.1) so we omit.
Let |${\mathcal{A}}=\kappa (\lambda ^{\alpha },g,\mu ^{\beta })$| for |$\lambda ^{\alpha },\mu ^{\beta } \in \Lambda _{\textbf{D}}, g\in \mathscr{D}_{\lambda ^{\alpha }\mu ^{\beta }}$|. Then |$x_{\lambda ^{\alpha }} T_{g} x_{\mu ^{\beta }} = [A]^!_{\mathfrak{d}} \, e_{\mathcal{A}}(x_{\mu ^{\beta }}).$|
Let |${\mathcal{B}} = \kappa (\lambda ^{\alpha },g_1,\mu ^{\beta })$| and |${\mathcal{A}} = \kappa (\mu ^{\beta },g_2,\nu ^{\gamma })$|, where |$\lambda ^{\alpha },\mu ^{\beta }, \nu ^{\gamma } \in \Lambda _{\textbf{D}}$|, |$g_1 \in \mathscr{D}_{\lambda ^{\alpha }\mu ^{\beta }}$|, and |$g_2 \in \mathscr{D}_{\mu ^{\beta }\nu ^{\gamma }}$|. Write |$\delta = \delta ({\mathcal{A}})$|. Then we have |$e_{\mathcal{B}} e_{\mathcal{A}}(x_{\nu ^{\gamma }}) = \frac{1}{[A]^!_{\mathfrak{d}}} x_{\lambda ^{\alpha }} T_{g_1} T_{(\mathscr{D}_{\delta } \cap W_{\mu ^{\beta }})g_2} x_{\nu ^{\gamma }}.$|
In the multiplication formulas below, we regard |$e_{\mathcal{A}}=0$| if |${\mathcal{A}}\not \in \Xi _{\textbf{D}}$|.
Suppose that |${\mathcal{A}}=A^{\textrm{sgn}({\mathcal{A}})}, {\mathcal{B}}, {\mathcal{C}}\in \Xi _{\textbf{D}}$|, and |$h\in [1,n]$|. Let |$\Gamma _r=\{t=(t_i)_{-n\leqslant i\leqslant n}\in{\mathbb{N}}^N | \sum _{i=-n}^nt_i=r\}.$|
- (1) If |$h\neq 1$|, |${\mathcal{B}}-rE_{h,h-1}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{B}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{B}})=s_l({\mathcal{A}})$|, thenwhere |$ {{\mathcal{A}}}_{t,h}=(A+t_pE_{h,p}^{\theta }-t_pE_{h-1,p}^{\theta },\textrm{sgn}(s_l({\mathcal{B}}),s_r({\mathcal{A}})))$|, |$s_l( {{\mathcal{A}}}_{t,h})=s_l({\mathcal{B}})$| and |$s_r( {{\mathcal{A}}}_{t,h})=s_r({\mathcal{A}})$|.(A.5.1)$$\begin{equation} e_{\mathcal{B}} e_{\mathcal{A}}=\sum_{t\in\Gamma_r}v^{2\sum_{k<p}t_pa_{h,k}}\prod_{p=-n}^{n}\left[\begin{array}{@{}cc@{}}a_{h,p}+t_p\\t_p\end{array}\right]e_{ {{\mathcal{A}}}_{t,h}}, \end{equation}$$
- (2) If |${\mathcal{B}}-rE_{1,0}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{B}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{B}})=s_l({\mathcal{A}})$|, then(A.5.2)$$\begin{align} e_{\mathcal{B}} e_{\mathcal{A}}=& \sum_{t\in\Gamma_r}v^{2\sum_{k<p}t_pa_{1,k}}(1+(1-\delta_{r,\frac{1}{2}\textrm{row}({\mathcal{A}})_0})(1-\delta_{a^{\prime}_{0,0},0})\delta_{a^{\prime}_{0,0},t_0})\nonumber\\&\prod_{p=-n}^{n}\left[\begin{array}{@{}cc@{}}a_{1,p}+t_p\\t_p\end{array}\!\!\!\right]e_{ {{\mathcal{A}}}_{t,1}}. \end{align}$$
- (3) If |$h\neq 1$|, |${\mathcal{C}}-rE_{h-1,h}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{C}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{C}})=s_l({\mathcal{A}})$|, thenwhere |$\widehat{{\mathcal{A}}}_{t,h}=(A-t_pE_{h,p}^{\theta }+t_pE_{h-1,p}^{\theta },\textrm{sgn}(s_l({\mathcal{C}}),s_r({\mathcal{A}})))$|, |$s_l(\widehat{{\mathcal{A}}}_{t,h})=s_l({\mathcal{C}})$| and |$s_r(\widehat{{\mathcal{A}}}_{t,h})=s_r({\mathcal{A}})$|.(A.5.3)$$\begin{equation} e_{\mathcal{C}} e_{\mathcal{A}}=\sum_{t\in\Gamma_r}v^{2\sum_{k>p}t_pa_{h-1,k}}\prod_{p=-n}^{n}\left[\begin{array}{cc}a_{h-1,p}+t_p\\t_p\end{array}\right]e_{\widehat{{\mathcal{A}}}_{t,h}}, \end{equation}$$
- (4) If |${\mathcal{C}}-rE_{0,1}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{C}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{C}})=s_l({\mathcal{A}})$|, then(A.5.4)$$\begin{align} e_{\mathcal{C}} e_{\mathcal{A}}=&\sum_{t\in\Gamma_r}v^{2\sum_{k>p}a_{0,k}t_p+2\sum_{p<k<-p}t_pt_k+\sum_{p<0}t_p(t_p-1)}\nonumber\\&\frac{[a_{0,0}+2t_0]_{\mathfrak{d}}^!}{[a_{0,0}]_{\mathfrak{d}}^![t_0]!}\prod_{p=1}^n\frac{[a_{0,p}+t_p+t_{-p}]!}{[a_{0,p}]![t_p]![t_{-p}]!}e_{\widehat{{\mathcal{A}}}_{t,1}}. \end{align}$$
Take |$r=1$| in Proposition A.5.4, we have the following corollary.
Suppose that |${\mathcal{A}}=A^{\textrm{sgn}({\mathcal{A}})}, {\mathcal{B}}, {\mathcal{C}}\in \Xi _{\textbf{D}}$| and |$h\in [1,n]$|.
- (1) If |$h\neq 1$|, |${\mathcal{B}}-E_{h,h-1}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{B}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{B}})=s_l({\mathcal{A}})$|, thenwhere |${\mathcal{A}}_p=(A+E_{h,p}^{\theta }-E_{h-1,p}^{\theta },\textrm{sgn}(s_l({\mathcal{B}}),s_r({\mathcal{A}})))$|.(A.5.13)$$\begin{equation} e_{\mathcal{B}} e_{\mathcal{A}}=\sum_{p=-n}^nv^{2\sum_{k<p}a_{h,k}}[a_{h,p}+1]e_{{\mathcal{A}}_p}, \end{equation}$$
- (2) If |${\mathcal{B}}-E_{1,0}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{B}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{B}})=s_l({\mathcal{A}})$|, then(A.5.14)$$\begin{equation} e_{\mathcal{B}} e_{\mathcal{A}}=\sum_{p\neq0}v^{2\sum_{k<p}a_{1,k}}[a_{1,p}+1]e_{{\mathcal{A}}_p} +v^{2\sum_{k<0}a_{1,k}}(2-\delta_{2,\textrm{row}(A)_0})[a_{1,0}+1]e_{{\mathcal{A}}_0}. \end{equation}$$
- (3) If |$h\neq 1$|, |${\mathcal{C}}-E_{h-1,h}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{C}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{C}})=s_l({\mathcal{A}})$|, thenwhere |${\mathcal{A}}(h,p)=(A-E_{h,p}^{\theta }+E_{h-1,p}^{\theta },\textrm{sgn}(s_l({\mathcal{C}}),s_r({\mathcal{A}})))$|.(A.5.15)$$\begin{equation} e_{\mathcal{C}} e_{\mathcal{A}}=\sum_{p=-n}^nv^{2\sum_{k>p}a_{h-1,k}}[a_{h-1,p}+1]e_{{\mathcal{A}}(h,p)}, \end{equation}$$
- (4) If |${\mathcal{C}}-E_{0,1}^{\theta }$| is diagonal, |$\textrm{col}({\mathcal{C}})=\textrm{row}({\mathcal{A}})$|, and |$s_r({\mathcal{C}})=s_l({\mathcal{A}})$|, then(A.5.16)$$\begin{align} e_{\mathcal{C}} e_{\mathcal{A}}=&\sum_{p\neq0}v^{2\sum_{k>p}a_{0,k}}[a_{0,p}+1]e_{{\mathcal{A}}(1,p)} +v^{2\sum_{k>0}a_{0,k}}\nonumber\\& \left([a_{0,0}+1]+(1-\delta_{0,a_{0,0}})v^{a_{0,0}}\right)e_{{\mathcal{A}}(1,0)}. \end{align}$$
An immediate application of the multiplication formulas is to demonstrate a stabilization property for |$\{\textbf{S} _{n,d}\mid d\in{\mathbb{N}}\}$| and further construct an algebra |$\mathcal{K}_n$| so that the multiplication rules on |$\mathcal{K}_n$| are compatible with the rules on any |$\textbf{S} _{n,d}$|. The algebras |$\mathcal{K}_n$| have been introduced by Fan and Li in loc. cit.
A.6 Schur duality
Let |$\mathfrak{g}$| be the simple Lie algebra of type |$\textbf{D}_d$|, and let |$\rho $| be the half sum of the positive roots of |$\mathfrak{g}$|. It was mentioned in a framework [25] that |$\Lambda _{\textbf{D}}$| can be viewed as the set of orbits of |$W$| on a (truncated) |$\rho $|-shifted weight lattice of |$\mathfrak{g}$|. Then the |$v$|-tensor space |$\bigoplus _{\lambda ^{\alpha }\in \Lambda _{\textbf{D}}}x_{\lambda ^{\alpha }}\textbf{H}$| can be viewed as the quantum version of the Grothendieck groups of the category |$\mathcal{O}$| of |$\mathfrak{g}$|-modules.
A Schur duality is also obtained in loc. cit. for each pair |$(\textbf{S} _f, \textbf{H})$| on the tensor space |$\mathop{\oplus }_{\lambda \in \Lambda _f} x_{\lambda } \textbf{H}$|.
If |$\Lambda _f=\Lambda ^+\sqcup \Lambda ^-$|, then |$\textbf{S} _f$| is the algebra |$\mathcal{S}^m$| in [14, §6.1]. The stabilization procedure affords a different quantum algebra |$\mathcal{K}^m$| in loc. cit.
Fan and Li told the authors in private conversations that they have also been aware of the Schur algebra |$\textbf{S} _f$| and the related Schur duality for |$\Lambda _f=\Lambda ^+$| or |$\Lambda ^0\sqcup \Lambda ^+$| although they did not write it down.
Funding
This work was supported by the Science and Technology Commission of Shanghai Municipality (18dz2271000 to L.L.) and the National Nature Science Foundation of China (11671108, 11871214 to L.L.).
Acknowledgments
The authors thank Huanchen Bao and Weiqiang Wang for helpful discussions. We also thank Catherina Stroppel for bringing [13] to our attention. We thank the referees for detailed comments on a previous version of the manuscript.
Communicated by Prof. Weiqiang Wang
References
