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To display a typeset version of a journal * %_ * article easily, we suggest that you retrieve the article in DVI, * %_ * PostScript, or PDF format. * %_ ************************************************************************** % Author Package file for use with AMS-LaTeX 1.2 \controldates{25-JUN-2001,25-JUN-2001,25-JUN-2001,25-JUN-2001} \documentclass{era-l} \issueinfo{7}{08}{}{2001} \dateposted{June 26, 2001} \pagespan{54}{62} \PII{S 1079-6762(01)00094-4} \usepackage{amssymb} \copyrightinfo{2001}{American Mathematical Society} \newcommand{\Ka}{{K_1}} \newcommand{\Kb}{{K_2}} \newcommand{\Kaa}{K_{1}^{-1}} \newcommand{\Kbb}{K_{2}^{-1}} \newcommand{\Ki}{{K_i}} \newcommand{\KKa}{{\overline{K_1}}} \newcommand{\KKb}{{\overline{K_2}}} \newcommand{\vv}{v^{-1}} \newcommand{\K}{\begin{bmatrix}K_i\\b_1\end{bmatrix}\begin{bmatrix}K_j\\b_2\end{bmatrix}} \newcommand{\KK}[2]{\begin{bmatrix}K_i\\#1 \end{bmatrix}\begin{bmatrix}K_j\\#2\end{bmatrix}} \newcommand{\ea}{E_\alpha^{(a)}} \newcommand{\fc}{F_\alpha^{(c)}} \newcommand{\fa}{F_\alpha^{(a)}} \newcommand{\ec}{E_\alpha^{(c)}} \newcommand{\Ha}{{H_i}} \newcommand{\Hb}{{H_j}} \newcommand{\ep}{\varepsilon} \newcommand{\al}{\alpha} \newcommand{\be}{\beta} \newcommand{\Ea}{E_{\alpha}} \newcommand{\Eb}{E_{\beta}} \newcommand{\Eg}{E_{\gamma}} \newcommand{\Ep}{E_{\rho}} \newcommand{\Eij}{E_{ij}} \newcommand{\Ekl}{E_{kl}} \newcommand{\vbinom}[2]{\begin{bmatrix}#1\\#2\end{bmatrix}} \newcommand{\vbinomm}[3]{\begin{bmatrix}#1; #2\\#3\end{bmatrix}} \newcommand{\Lnd}{\Lambda (n,d)} \newcommand{\Ua}{\mathbf{U}_{\mathcal{A}}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Q}{{\mathbb Q}} \newcommand{\R}{{\mathbb R}} \newcommand{\C}{{\mathbb C}} \newcommand{\g}{{\mathfrak g}} \newcommand{\n}{{\mathfrak n}} \newcommand{\dist}{\operatorname{Dist}} \newcommand{\per}{\operatorname{per}} \newcommand{\cov}{\operatorname{cov}} \newcommand{\non}{\operatorname{non}} \newcommand{\cf}{\operatorname{cf}} \newcommand{\add}{\operatorname{add}} \newcommand{\End}{\operatorname{End}} \newcommand{\Ext}{\operatorname{Ext}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Tor}{\operatorname{Tor}} \newcommand{\ch}{\operatorname{ch}} \newcommand{\ind}{\operatorname{ind}} \newcommand{\coind}{\operatorname{Coind}} \newcommand{\res}{\operatorname{res}} \newcommand{\soc}{\operatorname{soc}} \newcommand{\rad}{\operatorname{rad}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Dist}{\operatorname{Dist}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\im}{\operatorname{im}} \newcommand{\GL}{{\sf GL}} \newcommand{\SL}{{\sf SL}} \newcommand{\gl}{\mathfrak{gl}} \newcommand{\mysl}{\mathfrak{sl}} \newcommand{\B}{\mathcal{B}} \newcommand{\divided}[2]{#1^{(#2)}} \newcommand{\ct}{\chi}%content% \newcommand{\ctl}{\ct_L} \newcommand{\ctr}{\ct_R} \newcommand{\sqbinom}[2]{\begin{bmatrix}#1\\#2\end{bmatrix}} \newcommand{\U}{\mathbf{U}} \newcommand{\myS}{\mathbf{S}} \newcommand{\A}{\mathcal{A}} \newtheorem{theorem}{Theorem}%[section] \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} %\numberwithin{equation}{section} \begin{document} \title{Generators and relations for Schur algebras} \author{Stephen Doty} \address{Department of Mathematics, Loyola University, Chicago, IL 60626} \email{doty@math.luc.edu} \author{Anthony Giaquinto} \address{Department of Mathematics, Loyola University, Chicago, IL 60626} \email{tonyg@math.luc.edu} \subjclass[2000]{Primary 16P10, 16S15; Secondary 17B35, 17B37} \date{April 8, 2001} \keywords{Schur algebras, finite-dimensional algebras, enveloping algebras, quantized enveloping algebras} \commby{Alexandre Kirillov} \begin{abstract} We obtain a presentation of Schur algebras (and $q$-Schur algebras) by generators and relations, one which is compatible with the usual presentation of the enveloping algebra (quantized enveloping algebra) corresponding to the Lie algebra $\gl_n$ of $n\times n$ matrices. We also find several new bases of Schur algebras. \end{abstract} \maketitle \section{Introduction}\label{sec:intro} The classical Schur algebra $S(n,d)$ (over $\Q$) may be defined as the algebra $\End_{\Sigma_d}(V^{\otimes d})$ of linear endomorphisms on the $d$th tensor power of an $n$-dimensional $\Q$-vector space $V$ commuting with the action of the symmetric group $\Sigma_d$, acting by permutation of the tensor places (see \cite{Green}). Schur algebras determine the polynomial representation theory of general linear groups, and they form an important class of quasi-hereditary algebras. We identify $V$ with $\Q^n$. Then $V_\Z=\Z^n$ is a lattice in $V$. One can define a $\Z$-order $S_\Z(n,d)$ (the integral Schur algebra) in $S(n,d)$ by setting $S_\Z(n,d) = \End_{\Sigma_d}(V_\Z^{\otimes d})$. For any field $K$, one then obtains the Schur algebra $S_K(n,d)$ over $K$ by setting $S_K(n,d)= S_\Z(n,d)\otimes_\Z K$. Moreover, $S_\Q(n,d) \cong S(n,d)$. In the quantum case one can replace $\Q$ by $\Q(v)$ ($v$ an indeterminate), $V$ by an $n$-dimensional $\Q(v)$-vector space, and $\Sigma_d$ by the corresponding Hecke algebra $\mathbf{H}(\Sigma_d)$. Then the resulting commuting algebra, $\myS(n,d)$, is known as the $q$-Schur algebra, or quantum Schur algebra. It appeared first in work of Dipper and James, and, independently, Jimbo. Beilinson, Lusztig, and MacPherson \cite{BLM} have given a geometric construction of $\myS(n,d)$ in terms of orbits of flags in vector spaces. (See also \cite{Du}.) In this situation $\Z$ is replaced by the ring $\A=\Z[v,v^{-1}]$, and there is a corresponding ``integral'' form $\myS_\A(n,d)$ in $\myS(n,d)$. The above quantized objects specialize to their classical versions when $v=1$. We have more detailed information about the Schur algebras in rank $1$; see \cite{DG} and \cite{DGquantum}. Proofs of the main results will appear in \cite{PSA}. \section{Serre's presentation of $U$}\label{sec:presentU} Let $\Phi=\{\varepsilon_i - \varepsilon_j \mid 1\le i \ne j \le n\}$ be the root system of type $A_{n-1}$, with simple roots $\Delta=\{ \varepsilon_i - \varepsilon_{i+1} \mid i=1, \dots, n-1 \}$, where $\{\varepsilon_1, \dots, \varepsilon_n\}$ denotes the standard basis of $\Z^n$. The corresponding set $\Phi^+$ of positive roots is the set $ \Phi^+ = \{ \varepsilon_i - \varepsilon_j \mid 1\le i < j \le n \}$. The abelian group $\Z^n$ has a bilinear form $(\ ,\ )$ given by $(\varepsilon_i,\varepsilon_j)=\delta_{ij}$ (Kronecker delta). We write $\alpha_j:= \varepsilon_j - \varepsilon_{j+1}$. The enveloping algebra $U=U(\gl_n)$ is the associative algebra (with $1$) on generators $e_i,f_i$ ($i=1,\dots,n-1$) and $H_i$ ($i=1,\dots,n$) with relations \begin{align} H_i H_j &= H_j H_i, \tag{R1}\label{R1} \\ e_i f_j - f_j e_i &= \delta_{ij}(H_j - H_{j+1}), \tag{R2}\label{R2}\\ {}H_i e_j - e_j H_i = (\varepsilon_i, \alpha_j) e_j&, \quad H_i f_j - f_j H_i = -(\varepsilon_i, \alpha_j) f_j, \tag{R3}\label{R3} \end{align} \begin{equation}\label{R4} \begin{aligned} {}&e_i^2 e_j - 2 e_i e_j e_i + e_j e_i^2 = 0 \quad (|i-j|=1),\\ &e_i e_j - e_j e_i = 0 \quad (\text{otherwise}), \end{aligned} \tag{R4} \end{equation} \begin{equation}\label{R5} \begin{aligned} {}&f_i^2 f_j - 2 f_i f_j f_i + f_j f_i^2 = 0 \quad (|i-j|=1),\\ &f_i f_j - f_j f_i = 0 \quad (\text{otherwise}). \end{aligned} \tag{R5} \end{equation} For $\alpha\in \Phi$, let $x_{\alpha}$ denote the corresponding root vector of $\gl_n$ viewed as an element of $U$. We have in particular $e_i=x_{\alpha_i}$ and $f_i=x_{-\alpha_i}.$ \section{The quantized enveloping algebra}\label{sec:que} The Drinfeld-Jimbo quantized enveloping algebra $\U=\U_v(\gl_n)$, by definition, is the $\Q(v)$-algebra with generators $E_i$, $F_i$ ($1\leq i\leq n-1$), $K_i^{\pm 1}$ ($1\leq i\leq n$) and relations \begin{align} & K_i K_j =K_j K_i, \qquad K_i K_i^{-1} = K_i^{-1}K_i =1, \tag{Q1}\label{Q1}\\ & E_iF_j - F_jE_i = \delta_{ij}\frac{K_iK_{i+1}^{-1}-K_i^{-1}K_{i+1}}{v-v^{-1}},\tag{Q2}\label{Q2}\\ & K_i E_j = v^{(\ep_i, \al_j)}E_jK_i, \qquad K_i F_j = v^{-(\ep_i, \al_j)}F_jK_i, \tag{Q3}\label{Q3} \end{align} \begin{equation}\label{Q4} \begin{aligned} {}& E_i^2E_j - (v+v^{-1})E_iE_jE_i +E_jE_i^2 =0 \quad(|i-j|=1), \\ {}& E_iE_j - E_i E_j = 0 \quad(\text{otherwise}), \end{aligned}\tag{Q4} \end{equation} \begin{equation} \begin{aligned}\label{Q5} {}& F_i^2F_j - (v+v^{-1})F_iF_jF_i +F_jF_i^2 =0 \quad(|i-j|=1), \\ {}& F_iF_j - F_jF_i = 0 \quad(\text{otherwise}). \end{aligned}\tag{Q5} \end{equation} For $\alpha\in \Phi^+$, let $E_{\alpha}$ and $F_{\alpha}$ be the positive and negative quantum root vectors of $\U$ as defined by Jimbo \cite{Jimbo}. We have in particular $E_i=E_{\alpha_i}$ and $F_i=F_{\alpha_i}.$ \section{Main results: classical case} \label{sec:main} We now give a precise statement of our main results in the classical case. The first result describes a presentation by generators and relations of the Schur algebra over the rational field $\Q$. This presentation is compatible with the usual presentation (see section \ref{sec:presentU}) of $U=U(\gl_n)$. \begin{theorem} \label{thm:present:S} Over $\Q$, the Schur algebra $S(n,d)$ is isomorphic to the associative algebra (with $1$) on the same generators as for $U$ subject to the same relations \eqref{R1}--\eqref{R5} as for $U$, together with the additional relations: \begin{align} & H_1 + H_2 + \cdots + H_n = d, \tag{R6}\label{R6} \\ & H_k(H_k-1)\cdots(H_k-d) = 0 \qquad(k = 1, \dots, n). \tag{R7}\label{R7} \end{align} \end{theorem} The next result gives a basis for the Schur algebra which is the analogue of the Poincare-Birkhoff-Witt (PBW) basis of $U$. \begin{theorem} \label{thm:truncPBW} The algebra $S(n,d)$ has a ``truncated PBW'' basis (over $\Q$) which can be described as follows. Fix an integer $k_0$ with $1\le k_0 \le n$ and set \[ G=\{ x_\alpha \mid \alpha\in \Phi \} \cup \{H_k \mid k \in \{1,\dots,n\}-\{k_0\}\} \] and fix some ordering for this set. Then the set of all monomials in $G$ (with specified order) of total degree at most $d$ is a basis for $S(n,d)$. \end{theorem} Our next result constructs the integral Schur algebra $S_\Z(n,d)$ in terms of the generators given above. We need some more notation. For $B$ in $\N^n$, we write \[ H_B = \prod_{k=1}^n \dbinom{H_k}{b_k}. \] Let $\Lambda(n,d)$ be the subset of $\N^n$ consisting of those $\lambda \in \N^n$ satisfying $|\lambda|=d$; this is the set of $n$-part compositions of $d$. Given $\lambda \in \Lambda(n,d)$ we set $1_\lambda:=H_{\lambda}.$ One can show that the collection $\{1_\lambda\}$ as $\lambda$ varies over $\Lambda(n,d)$ forms a set of pairwise orthogonal idempotents in $S_\Z(n,d)$ which sum to the identity element. For $m\in \N$ and $\alpha \in \Phi$, set $\divided{x_\alpha}{m}:= x_\alpha^m/(m!)$. Any product in $U$ of elements of the form \[ \divided{x_\alpha}{m}, \quad \dbinom{H_k}{m}\qquad (m\in \N, \alpha\in\Phi, k\in \{1,\dots, n\}), \] taken in any order, will be called a {\em Kostant monomial}. Note that the set of Kostant monomials is multiplicatively closed, and spans $U_\Z$. We define a function $\ct$ (content function) on Kostant monomials by setting \[ \ct(\divided{x_\alpha}{m}):= m\,\varepsilon_{\max(i,j)}, \quad \ct( \binom{H_k}{m} ) := 0, \] where $\alpha = \varepsilon_i - \varepsilon_j$ ($i\ne j$), and by declaring that $\ct(XY) = \ct(X)+\ct(Y)$ whenever $X,Y$ are Kostant monomials. We write any $A\in \N^{\Phi^+}$ in ``multi-index'' form $A=(a_\alpha)_{\alpha\in \Phi^+}$ and set $|A|:= \sum_{\alpha\in \Phi^+} a_\alpha$. For $A, C \in \N^{\Phi^+}$ we write \[ e_A = \prod_{\alpha\in \Phi^+} \divided{x_\alpha}{a_\alpha}, \quad f_C = \prod_{\alpha\in \Phi^-} \divided{x_{\alpha}}{c_\alpha} \] where the products in $e_A$ and $f_C$ are taken relative to any fixed orders on $\Phi^+$ and $\Phi^-$. \begin{theorem} \label{thm:idemp:basis} The integral Schur algebra $S_\Z(n,d)$ is the subring of $S(n,d)$ generated by all divided powers $ \divided{e_j}{m},\, \divided{f_j}{m} (j\in\{1,\ldots , n-1\}, m\in \N)$. Moreover, this algebra has a $\Z$-basis consisting of all \begin{equation}\label{B1} e_A 1_\lambda f_C \qquad (\ct(e_A f_C) \preceq \lambda)\tag{B1} \end{equation} and another such basis consisting of all \begin{equation}\label{B2} f_A 1_\lambda e_C \qquad (\ct(f_A e_C) \preceq \lambda)\tag{B2} \end{equation} where $A,C \in \N^{\Phi^+}$, $\lambda\in \Lambda(n,d)$, and where $\preceq$ denotes the {\em componentwise} partial ordering on $\N^n$. \end{theorem} Finally, we have another presentation of the Schur algebra by generators and relations. This presentation has the advantage that it possesses a quantization of the same form, in which we can set $v=1$ to recover the classical version. \begin{theorem} \label{thm:idemp:present} For each $i\in \{1,\dots,n-1\}$ write $\alpha_i := \varepsilon_i -\varepsilon_{i+1}$. The algebra $S(n,d)$ is the associative algebra (with $1$) given by generators $1_\lambda$ ($\lambda\in \Lambda(n,d)$), $e_i$, $f_i$ ($i\in \{1,\dots,n-1\}$) subject to the relations \begin{equation}\label{S1} \begin{gathered} 1_\lambda 1_\mu = \delta_{\lambda,\mu} 1_\lambda \quad(\lambda,\mu\in \Lambda(n,d)),\\ \sum_{\lambda\in \Lambda(n,d)} 1_\lambda = 1, \end{gathered}\tag{S1} \end{equation} \begin{equation}\label{S2} \begin{gathered} e_i 1_\lambda = \begin{cases} 1_{\lambda+\alpha_i} e_i & \text{if $\lambda+\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases}\\ f_i 1_\lambda = \begin{cases} 1_{\lambda-\alpha_i} f_i & \text{if $\lambda-\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases}\\ 1_\lambda e_i = \begin{cases} e_i 1_{\lambda-\alpha_i} & \text{if $\lambda-\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases}\\ 1_\lambda f_i = \begin{cases} f_i 1_{\lambda+\alpha_i} & \text{if $\lambda+\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases} \end{gathered}\tag{S2} \end{equation} \begin{equation}\label{S3} e_i f_j - f_j e_i = \delta_{ij} \sum_{\lambda\in \Lambda(n,d)} (\lambda_j-\lambda_{j+1}) 1_\lambda, \tag{S3} \end{equation} along with the Serre relations \eqref{R4}, \eqref{R5}, for $i,j \in \{1, \dots,n-1\}$. \end{theorem} \section{Main results: quantized case} \label{qsec:main} Our main results in the quantized case are similar in form to those in the classical case. The first result describes a presentation by generators and relations of the quantized Schur algebra over the rational function field $\Q(v)$. This presentation is compatible with the usual presentation (see section \ref{sec:que}) of $\U=\U_v(\gl_n)$. \setcounter{theorem}{0} \renewcommand{\thetheorem}{\arabic{theorem}$^\prime$} \begin{theorem} \label{qthm:present:S} Over $\Q(v)$, the $q$-Schur algebra $\myS(n,d)$ is isomorphic with the associative algebra (with $1$) on the same generators as for $\U$ subject to the same relations \eqref{Q1}--\eqref{Q5} as for $\U$, together with the additional relations: \begin{align} & K_1 K_2 \cdots K_n = v^d \tag{Q6}\label{Q6}, \\ & (K_j-1)(K_j-v)(K_j-v^2)\cdots(K_j-v^d) = 0 \qquad(j=1, \dots, n). \tag{Q7}\label{Q7} \end{align} \end{theorem} The next result gives a basis for the $q$-Schur algebra which is the analogue of the Poincare-Birkhoff-Witt (PBW) type basis of $\U$, given in Lusztig \cite[Prop.\ 1.13]{Lusztigbook}. \begin{theorem} \label{qthm:truncPBW} The algebra $\myS(n,d)$ has a ``truncated PBW type'' basis which can be described as follows. Fix an integer $k_0$ with $1\le k_0 \le n$ and set \[ G'=\{ E_\alpha, F_{\alpha} \mid \alpha\in \Phi^+ \} \cup \{K_k \mid k \in \{1,\dots,n\}-\{k_0\} \} \] and fix some ordering for this set. Then the set of all monomials in $G'$ (with specified order) of total degree at most $d$ is a basis for $\myS(n,d)$. \end{theorem} Note that setting $v=1$ in the basis of Theorem \ref{qthm:truncPBW} does not yield the basis of Theorem \ref{thm:truncPBW} since $K_i$ acts as the identity when $v=1$. Our next result constructs the integral $q$-Schur algebra $\myS_\A(n,d)$ in terms of the generators given above. For $B$ in $\N^n$, we write \[ K_B = \prod_{j=1}^n \sqbinom{K_j}{b_j}, \] where for indeterminates $X,X^{-1}$ satisfying $X X^{-1} = X^{-1} X = 1$ and any $t\in \N$ we formally set \[ \sqbinom{X}{t}:=\prod_{s=1}^{t} \frac{Xv^{-s+1}-X^{-1}v^{s-1}}{v^s-v^{-s}}, \] an expression that makes sense if $X$ is replaced by any invertible element of a $\Q(v)$-algebra. Given $\lambda \in \Lambda(n,d)$ we set (when we are in the quantum case) $1_\lambda:= K_\lambda$. It turns out that, just as in the classical case, the collection $\{1_\lambda\}$ as $\lambda$ varies over $\Lambda(n,d)$ forms a set of pairwise orthogonal idempotents in $\myS_\A(n,d)$ which sum to the identity element. Let $[m]$ denote the quantum integer $[m]:= (v^m-v^{-m})/(v-v^{-1})$ and set $[m]! := [m][m-1] \cdots [1]$. Then the $q$-analogues of the divided powers of root vectors are defined to be $ \divided{E_\alpha}{m}:= E_\alpha/[m]!$ and $\divided{F_\alpha}{m}:= F_\alpha/[m]!.$ Any product in $\U$ of elements of the form \[ \divided{E_\alpha}{m}, \quad \divided{F_\alpha}{m}, \quad K_j^{\pm 1}, \quad\sqbinom{K_j}{m} \qquad (m\in \N,\ \alpha\in\Phi,\ j\in \{1,\dots, n\}), \] taken in any order, will be called a {\em Kostant monomial}. As before, the set of Kostant monomials is multiplicatively closed, and spans $\U_\A$. The definition of content $\ct$ of a Kostant monomial is obtained similarly, by setting \[ \ct(\divided{E_\alpha}{m}) = \ct(\divided{F_\alpha}{m}) := m\,\varepsilon_{\max(i,j)}, \quad \ct(K_l^{\pm 1}) = \ct( \sqbinom{K_l}{m} ) := 0 \] where $\alpha = \varepsilon_i - \varepsilon_j \in \Phi^+$, and by declaring that $\ct(XY) = \ct(X)+\ct(Y)$ whenever $X,Y$ are Kostant monomials. For $A, C \in \N^{\Phi^+}$ we write \[ E_A = \prod_{\alpha\in \Phi^+} \divided{E_\alpha}{a_\alpha}, \quad F_C = \prod_{\alpha\in \Phi^+} \divided{F_{\alpha}}{c_\alpha} \] where the products in $E_A$ and $F_C$ are taken relative to any two specified orderings on $\Phi^+$. \begin{theorem} \label{qthm:idemp:basis} The integral $q$-Schur algebra $\myS_\A(n,d)$ is the subring of $\myS(n,d)$ generated by all quantum divided powers $ \divided{E_j}{m},\, \divided{F_j}{m} \,(j\in \{ 1,\ldots , n-1\}, m\in \N),$ along with the elements $\sqbinom{K_j}{m}$ ($j \in \{1,\dots,n\}$, $m\in \N$). Moreover, this algebra has a basis over $\A$ consisting of all \begin{equation}\label{B1'} E_A 1_\lambda F_C \qquad (\ct(e_A f_C) \preceq \lambda)\tag{B$1^\prime$} \end{equation} and another such basis consisting of all \begin{equation}\label{B2'} F_A 1_\lambda E_C \qquad (\ct(f_A e_C) \preceq \lambda)\tag{B$2^\prime$} \end{equation} where $A,C \in \N^{\Phi^+}$, $\lambda\in \Lambda(n,d)$, and where $\preceq$ denotes the {\em componentwise} partial ordering on $\N^n$. \end{theorem} Unlike the truncated PBW basis, the bases of Theorem \ref{qthm:idemp:basis} do specialize when $v=1$ to their classical analogues given in Theorem \ref{thm:idemp:basis}. Finally, we have another presentation of the $q$-Schur algebra by generators and relations. This presentation has the advantage that by setting $v=1$, we recover the classical version given in Theorem \ref{thm:idemp:present}. The relations of the presentation are similar to relations that hold for Lusztig's modified form $\dot{\U}$ of $\U$. (See \cite[Chap.\ 23]{Lusztigbook}.) \begin{theorem} \label{qthm:idemp:present} For each $i\in \{1,\dots,n-1\}$ write $\alpha_i := \varepsilon_i -\varepsilon_{i+1}$. The algebra $\myS(n,d)$ is the associative algebra (with $1$) given by generators $1_\lambda$ ($\lambda\in \Lambda(n,d)$), $E_i$, $F_i$ ($i\in \{1,\dots,n-1\}$) subject to the relations \begin{equation}\label{S1'} \begin{gathered} 1_\lambda 1_\mu = \delta_{\lambda,\mu} 1_\lambda \quad(\lambda,\mu\in \Lambda(n,d)),\\ \sum_{\lambda\in \Lambda(n,d)} 1_\lambda = 1, \end{gathered} \tag{S$1^\prime$} \end{equation} \begin{equation}\label{S2'} \begin{gathered} E_i 1_\lambda = \begin{cases} 1_{\lambda+\alpha_i} E_i & \text{if $\lambda+\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases}\\ F_i 1_\lambda = \begin{cases} 1_{\lambda-\alpha_i} F_i & \text{if $\lambda-\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases}\\ 1_\lambda E_i = \begin{cases} E_i 1_{\lambda-\alpha_i} & \text{if $\lambda-\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases}\\ 1_\lambda F_i = \begin{cases} F_i 1_{\lambda+\alpha_i} & \text{if $\lambda+\alpha_i \in \Lambda(n,d)$},\\ 0 & \text{otherwise}, \end{cases} \end{gathered}\tag{S$2^\prime$} \end{equation} \begin{equation} \label{S3'} E_i F_j - F_j E_i = \delta_{ij} \sum_{\lambda\in \Lambda(n,d)} [\lambda_j-\lambda_{j+1}] 1_\lambda, \tag{S$3^\prime$} \end{equation} along with the $q$-Serre relations \eqref{Q4}, \eqref{Q5}, for $i,j \in \{1, \dots,n-1\}$. \end{theorem} \section{Other results}\label{sec:other} \subsection{Triangular decomposition.} One can define the plus, minus, and zero parts of Schur algebras in terms of the generators, as follows. (These subalgebras have been studied before.) The plus part $S^+(n,d)$ (resp., the minus part $S^-(n,d)$) is the subalgebra of $S(n,d)$ generated by all $x_\alpha$, $\alpha\in \Phi^+$ (resp., $\alpha\in \Phi^-$). The zero part $S^0(n,d)$ is the subalgebra generated by all $H_j$, $j=1,\dots, n$. We also have the Borel Schur algebras $S^{\geqslant0}(n,d)$ (resp., $S^{\leqslant0}(n,d)$), the subalgebra generated by $S^+(n,d)$ (resp., $S^-(n,d)$) together with $S^0(n,d)$. The appellations $S_\Z^+(n,d)$, $S_\Z^-(n,d)$, $S_\Z^0(n,d)$, $S_\Z^{\geqslant0}(n,d)$, $S_\Z^{\leqslant0}(n,d)$ will denote the intersection of the appropriate algebra from above with the integral form $S_\Z(n,d)$. The algebra $S=S(n,d)$ has a triangular decomposition $S=S^+S^0S^-$. We show that in this decomposition one can permute the three factors in any order. Moreover, the same result holds over $\Z$. The zero part $S_\Z^0(n,d)$ is the algebra generated by all $\binom{H_j}{m}$ for $j=1,\dots,n$ and $m \in \N$. We give in \cite{PSA} a presentation of $S^0(n,d)$ by generators and relations. In particular, we prove that $H_B = 0$ whenever $|B| > d$ ($B\in \N^n$) and that the set of (pairwise orthogonal) idempotents $1_\lambda$, $\lambda\in \Lambda(n,d)$, is a $\Z$-basis of $S_\Z^0(n,d)$. $S_\Z^+(n,d)$ (resp., $S_\Z^-(n,d)$) is the algebra generated by all divided powers $\divided{x_\alpha}{m}$ for $\alpha\in \Phi^+$ (resp., $\alpha\in \Phi^-$) and $m\in \N$. It is an easy consequence of the commutation formulas \eqref{S2} that each generator $x_\alpha$ is nilpotent of index $d+1$; see \cite{PSA} for details. Moreover, from our main results we see easily that the set of all $e_A$ (resp., $f_A$) such that $|A|\le d$ is a $\Z$-basis for the algebra $S_\Z^+(n,d)$ (resp., $S_\Z^-(n,d)$). It also follows immediately from our results that the set of all $e_A 1_\lambda$ (resp., $1_\lambda f_A$) satisfying $\ct(e_A)\preceq \lambda$ is an integral basis for the Borel Schur algebra $S_\Z^{\geqslant0}(n,d)$ (resp., $S_\Z^{\leqslant0}(n,d)$). Similar statements to the above hold in the quantum case. In particular, as a corollary of the commutation formulas \eqref{S2'} one can give a simple proof of \cite[Prop.\ 2.3]{RGreen}. \subsection{Explicit reduction formulas} Fix a positive root $\alpha$ and write $\alpha = \varepsilon_i - \varepsilon_j$ for $i2$. We also have from \cite{DGquantum} the following $q$-version of the above, which presents the $q$-Schur algebra $\myS(2,d)$ as a quotient of the quantized enveloping algebra $\U(\mysl_2)$. \setcounter{theorem}{4} \renewcommand{\thetheorem}{\arabic{theorem}$^\prime$} \begin{theorem} Over $\Q(v)$, the quantum Schur algebra $\myS(2,d)$ is isomorphic to the algebra generated by $E$, $F$, $K^{\pm 1}$ subject to the relations: \begin{align*} & KK^{-1}=K^{-1}K=1,\\ &K E K^{- 1} =v^{2}E, \qquad K F K^{- 1} =v^{- 2}F, \\ & EF-FE=\frac{K - K^{-1}}{v-v^{-1}}, \\ & (K-v^d)(K-v^{d-2})\cdots (K-v^{-d+2})(K-v^{-d})=0. \end{align*} \end{theorem} \subsection{Hecke algebras.} Suppose that $n\ge d$. Let $\omega = (1^d)$. Then the subalgebra $1_\omega \myS(n,d) 1_\omega$ is isomorphic with the Hecke algebra $\mathbf{H}(\Sigma_d)$. The nonzero elements of the basis \eqref{B1'} of the form $1_\omega E_A F_C 1_\omega$ form a basis of the Hecke algebra; similarly for elements of the basis \eqref{B2'} of the form $1_\omega F_A E_C 1_\omega$. Moreover, taking $d=n$, we can see that $\mathbf{H}=\mathbf{H}(\Sigma_n)$ is generated by the elements $1_\omega E_i F_i 1_\omega$ ($1 \le i \le n-1$). Alternatively, $\mathbf{H}$ is generated by the $1_\omega F_i E_i 1_\omega$ ($1 \le i \le n-1$). One can easily describe the relations on these generators, thus obtaining a presentation of $\mathbf{H}$ which is closely related to one in \cite{Wenzl}. \bibliographystyle{amsplain} \begin{thebibliography}{10} \bibitem{BLM} A.A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\GL_n$, {\em Duke Math. J.} {\bf61} (1990), 655--677. \MR{91m:17012} \bibitem{DG} S. Doty and A. 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