数理解析研究所講究録
第 1795 巻 2012 年 45-58

45

Some developments on Schur functors and dominant dimension
Ming Fang*

Institute of Mathelnatics
Chinese Academy of Sciences, Beijing, P.R.China, 100190
Email: fnling@amss.ac.cn

Abstract

Schur functors have been playing crucial roles in representation theory to relate different
algebras. Prominent examples include the remarkable Schur-Weyl duality which relates Schuralgebras to group algebras of symmetric groups and Soergel’s struktursatz which relates
blocks of categoryof a complex semisimple Lie algebra to coinvariant algebras of thecorresponding flag manifold. This survey contains our recent investigation on homologicalproperties of Schur functors in terms of dominant dimension as well as a number of new
applications of dominant dimension in representation theory.
$\mathcal{O}$

Keywords Schur functor. Dominant dimension, Schur algebraMR(2010) Subject Classification $16G10,13E10$

1Introduction

General linear and symmetric groups are two fundamental objects in group theory. A hundredyears ago, I. Schur determined the polynomial representations of the complex general linear
group
in his doctoral dissertation. An essential part of Schur idea was to set up a
correspondence between representatioms ofof fixed homogenous degree , and representations of the finite symmetric groupon letters, and through this correspondence to applyC.
. Such an idea was later publicized by H.
discovery of the characters of
Weyl and J. A. Green in the well-known Schur Weyl duality, and seemed to appear frequently
in many other contexts, like Bernstein, Gelfand and Gelfand theory on projective functors;
Soergel is struktursatz which relates blocks of category of a complex semisimple Lie algebra
to coinvariant algebras alid the theory of KZ functors which relate the categoryof rational
cherednik algebras to Hecke algebras. Nevertheless, Schur original idea on relating representations of general linear and symmetric groups is vely much subtle in modular situations since
the correspondence above (via the Schur functor) is no longer an equivalence and cohomologies doubtlessly come in to play. As far as we have understood, there $aJe$ two approaches torescue Schur’s idea, one by Hemmer, Kleshchev and Nakano on cohomologies of Schur functors$GL_{n}(\mathbb{C})$

$s$

$r$

$GL_{n}(\mathbb{C})$$\Sigma_{r}$$\mathbb{R}obenius^{:}s$$r$$\Sigma_{r}[14]$$s$$\mathcal{O}$$\mathcal{O}$$s$“Supported by the National Natural Science Foundation of China $(No.11001253)$ .

46[16, 19] and the other by Koenig, Slungard and Xi on double centralizer properties alid dominantdimension [21].

From the elelnentary representation theory of symmetric groups, we know that every Youngmodule of symmetric groups admits a filtration whose sub-quotients are isomorphic to Specht
modules, whereas the filtration multiplicities are not well-defined. As typical examples, we havethat when the characteristic of the field equals 2.
and when $p=3$ , there exists
a short exact sequence of Specht modules, see [22]$S^{(2,0)}\cong S^{(1,1)}$$p$$0arrow S^{(5,1,1)}arrow S^{(3,3,1)}arrow S^{(2,1^{5})}arrow 0$

However, based on Kleshchev and Nakano’s comparison on coholnologies of general linear andsymmetric groups in [19], Hemmer alld Nakano proved that when $p\geq 5$ , the Schur functor

preserves extension groups between Weyl modules up to degree $p-3$ and as a result, themultiplicities of Specht modules in Young modules are well-defined [16]. see also [17, 18] forsome further advalices of the salne spirit. In [5], Doty, Erdmann and Nakaiio set up a generalframework so as to generalize Hemmer, Kleshchev and Nakano’s work to a broad context withthe emphasis on cohomologies of Schur functors.
As the main ingredient in Koenig, Slungard and Xi approach [21], dominant dimension
was introduced to study QF-3 rings, one of the several generalizations of quasi-Frobenius rings,
by Tadtikawa and Morita [24, 25], see also [26, 28]. In representation theories, it behaves very
subtle and was mainly used to study torsionless modules and double centralizer properties, thecrucial point in Schur technique. In [21], it was firstly applied in algebraic Lie theory to givea computation-free proof of both Schur-Weyl duality and Soergel struktursatz by showing thedominant dimension of algebras in question at least 2.
$s$

$s$

$s$

In [7, 9], we combined two approaches above and considered the following setting. Let $(A, e)$be a QF-3 algebra wliere is an idempotent in such that $Ae$ is a basic projective. injective
and faithful A-module, see Section 2.4. Consider the Schur functor after [5, 14] afforded by
$A$

$e$

$e$

$f=eA\otimes_{A}-:A-mod arrow eAe$ -mod

Motivated by [16, 19], we shall assume that
$Hom_{eAe}(f(P), f(Q))$
$Q$

$f$

induces natural isomorphisms $Hom_{A}(P, Q)\cong$(called double centralizer property) for all projective A-modules $P$ and, just as in tlie (quaiitum) Schur-Weyl duality and Soergel$s$struktursatz, alid consider thefollowing questions(a) find maximal $n\in N$ such that $f$ induces an isomorphism $Ext_{A}^{i}(P, Q)\cong Ext_{eAe}^{i}(eP, eQ)$ for

any projective A-modules(b) for a full subcategory

$P$

and

$Q$

and $0\leq i\leq n$ .

of A-mod, find maximal
such that induces an isomorphism$Ext_{A}^{i}(M, N)\cong Ext_{eAe}^{i}(eM, eN)$ for any modules $M,$.
and$\mathscr{C}$$n(\mathscr{C})$$N\in \mathscr{C}$$f$$0\leq i\leq n(\mathscr{C})$It turns out that the solutions to these questions are hinted around dominant dimensions. Indeed,(a) has been settled by M\"uller in terms of dominalit dimension of the algebra [23], see also Section
2.4. As for (b), we will restrict ourselves to consider, say,for quasi-hereditary algebras.
$\ovalbox{\tt\small REJECT}(\triangle)$

47
In particular, Kleshchev and Nakano’s result for classical Schur algebras $S_{k}(n, r)$ can be restated,as: assume that $n\geq r$ and is an infinite field of characteristic $p\geq 5$ , then$S_{k}(n,r)$
whereconsists of those-modules which are filtered by Weyl modules. Hemmer andNakano result for the quantized Schur algebra $S_{q}(n, r)$ can be restated as: assume that $n\geq r$
forand is an -th root of unity with
. In [7], we computed, thensome specific cases. n\‘amely $n(\mathscr{G}(\Delta))=p-3$ for $S_{k}(p,p)$ where $p>0$ is the characteristic of the
field $k;n(\mathscr{G}(\Delta))=0$ for any non-semisimple block algebra of the Berlistein-Gelfand-Gelfand
category of a complex semisimple Lie algebra. Furthermore we proved a positive correlation
betweenand the dominant dimension of a QF-3 standardly stratified algebra, i.e., thelarger the dominant dimension of the algebra, the larger.
$k$$n(\ovalbox{\tt\small REJECT}(\Delta))\geq p-3$$\mathscr{G}(\Delta)$$s$$\ell\geq 4$$\ell$$q$$n(\ovalbox{\tt\small REJECT}(\triangle))$$n(\ovalbox{\tt\small REJECT}(\Delta))\geq 1$

$A$

$O$

$n(\mathscr{G}(\triangle))$$n(\mathscr{G}(\Delta))$by
In [9], we obtained a quantitative relation between dominant dimensions andrestricting to a subclass of quasi-hereditary algebras, which includes (quantized, cyclotomic)
Schur algebras, block algebras of BGG category as basic examples. As main results, we provedthat for each algebra in this class, its dominant dimension dom. $\dim$ is exactly twice thatof the characteristic tilting module, hence an even number; the Ringel-dual of has dominantdilnension the same as
dom. $\dim A/2-2$ where
dom. $\dim A-2$ andis the full subcategory of finitely generated projective A-modules: characterized how dominantdimension changes under trumcation process; computed the dolninant dimension of classicalSchur algebras $S_{k}(n, r)$ for $n\geq r$ . As a direct application, we recovered alud strengthened theresults above by Hemmer, Kleshchev and Nakano, see also Section 3 and Section 5, reproved andgeneralized James and Donkin result on the torsionless property of (quantum) Weyl modules.
$n(\mathscr{P}(\Delta))$

$A$

$A$

$A$

$A;n(\mathcal{P}_{A})=$$n(\ovalbox{\tt\small REJECT}(\Delta))=$$\mathcal{P}_{A}$$s$In [10. 11, 12], we aimed to develop new effective characterizations of dominalit dimension as
well as more applications. To do so, we enlarged the algebra class in [9] to introduce the so-called
generalized symmetrc algebras. By definition, this is an algebra class consists of endomorphism
algebras of generators over symmetric algebras. It contains some quasi-hereditary algebras(finite global dimension) like Schur algebras, as well as all symmetric algebras (generally infinite
global dimension) like Hecke algebras. To be precise, in [10], we obtained some equivalentcharacterizations of generalized symmetric algebras and a refinement of the Tachikawa-Morita
correspondence. In particular, we extended a characterization of dominant dimension in [9] togeneralized symmetric algebras. In [11], we made use of the property that every generalized
symmetric algebra
admits a special coproduct (usually without a counit) to construct a
Hochschild cocomplex and proved that the highest degree where the cocomplex is exact at each
degree below, is dom. $\dim A+2$ . As a consequence. the computation of dominant dimensions
in certain cases will be accessible to computer programmes once the above coproduct maps are
explicitly known. As another application, we obtained for each generalized $sy$lnlnetric algebra ,the isomorphisms between the i-th Hochschild cohomologies of and its centralizer subalgebra$eAe$ for $0\leq i\leq$ dom. $\dim A-2$ . In [12], we gave an explicit construction of a multiplication on
the dual space $A_{k}(n, r)$ of $S_{k}(n, r)$ and proved that tlie image of this multiplication lnap coincides
with the Doty coalgebra $D_{k}(n, r)$ . Moreover we proved that $D_{k}(n.r)=A_{k}(n, r)$ if and only if$r\leq n(p-1)$ . By use of this multiplication map, we proved that $S_{k}(n, r)$ is generalized symmetricwhen $r\leq n(p-1)$ , where is the characteristic of the field and the dual of this multiplication
map is a prerequisite coproduct map in [10]. Combined with the strategy developed in [10], we
$A$

$A$

$A$

$p$

48
are able to compute the dominant dimension of these Schur algebras in principle.
This paper originates from the talk on the workshop”Topics in Combinatorial Representation
Theory” (Kyoto, Oct. 2011) given by the author. To make it more accessible, we will include in
Section 1 some basic theory on representations of general linear and symmetric groups, dominantdimensions etc. and omit all proofs to keep the size reasonable. The interested readers are invited
to refer to another survey “Dominant dilnension and aJmost relatively true versions of Schurtheorem’ given by Koenig from a somewhat different point of view [20].
$s$Acknowledgement. The author thanks Professor S. Ariki for greatly supporting his one
year visit in Osaka University and Professor R. Sakamoto for the generous financial support on
attending the workshop.

2

Preliminaries

In this section, we fix the notation and recall some basic theory of representations of generallinear groups and domil$ial$) dimension, etc.
$t$

Representations of general linear and symmetric group

2.1

Let and be natural numbers. Define $I(n, r)$ to be the set of sequencesof
integers withof integers suchthe set of sequences
andthat. Note tliat theand $\lambda_{1}+\cdots+\lambda_{n}=r$ . Letsymmetricon r-letters has a natural right action on $I(n, r)$ by place permutations:. We define
if they belong to the same -orbit
forand define Stab to be the stabilizer subgroup of in . Elements in $I(n, r)$ are called multiindices and elements inare called partitions of . A partition is called l-restricted forsome number if $0\leq\lambda_{i}-\lambda_{i+1}\leq l-1$ for all .
$n$$\underline{i}=(i_{1}, \ldots, i_{r})$$r$$i_{\rho}\in\{1, \ldots, n\}$$\Lambda^{+}(n, r)$$\lambda=(\lambda_{1}\ldots.

$\lambda_{1}\geq\cdots\geq\lambda_{n}\geq 0$, \lambda_{n})$$\Lambda^{+}(r)=\bigcup_{n\geq 1}\Lambda^{+}(n, r)$$\Leftrightarrow\infty loup\Sigma_{r}$$\sigma\in\Sigma_{r}$$(i_{1}, \ldots, i_{r})\cdot\sigma=(i_{1\sigma}, \ldots, i_{r\sigma})$$\Sigma_{r}$$\underline{i}\sim\underline{j}$$(\underline{i})$$\underline{i}$$\Lambda^{+}(r)$$\Sigma_{r}$$\lambda$

$r$

$i$

$\ell$

Let be an infinite field of characteristic $p\geq 0$ . For each partition of , letdenote theSpecht module andthe dual Specht module indexed by . It is well-known that if $p=0$ , then-modules; ifandis a complete set of non-isomorphic irreducible$p>0$ , then for each p-restricted partition , the dual Specht module
has a unique irreducible
quotient which we denoted by. Furthermore
forms a complete
set of pairwise non-isomorphic irreducible
be the Young module indexed-modules. Let
by . For more information about representations of symmetric groups, see [13] or [14, 22].
$k$$\lambda$$S^{\lambda}$$\lambda$$S_{\lambda}$$S_{\lambda}\cong S^{\lambda}$$r$$\{S_{\lambda}|\lambda\in\Lambda^{+}(r)\}$$\Sigma_{r}$$\lambda$$S_{\lambda}$$\{D_{\lambda}|\lambda\in\Lambda^{+}(r)_{l}\succ restricted\}$$D_{\lambda}$$Y^{\lambda}$$k\Sigma_{r}$$\lambda$

Let $E$ be an -dilnelisional k-vector space. Then the r-tensor spacerightaction by place permutation, namely$n$$E^{O-r}$admits a natural$\Sigma_{r}$$e_{i_{1}}\otimes\cdots\otimes e_{i_{r}}\cdot\sigma=e_{i_{1\sigma}}\otimes\cdots\otimes e_{i_{r\sigma}}$,$i_{1},$$\ldots,$$i_{r}\in\{1\ldots., n\},$ $\sigma\in\Sigma_{r}$Definition 2.1. The Schur algebra $S_{k}(n, r)$ is defined to be the endomorphism algebra
Remark. (1) If we replace the group algebra$k\Sigma_{r}$$End_{\Sigma_{f}}(E^{\otimes r})$by its quantization, the Hecke algebra$\mathcal{H}_{q}(r)$

.

49
and

$E^{\prime p_{r}}$by the q-tensor space in the definition above, we will get the q-Schur algebras $S_{q}(n, r)$ .

For more details about Schur algebras or q-Schur algebras, see [4, 14, 22].
(2) Sometimes, it is convenient to work with the following equivalent definition ( $w1$ 1

admits a quantized analogue, see [4] . Let A. $(n)$ be the polynomial ring in$)$and $A_{k}(n, r)$ be the r-th homogenous subspace.
with the comultiplicationand counit defined by$\{c_{\tau,j}|1\leq i,j\leq n\}$$A_{\lambda}(n)$also-indeterminants
makes a bialgebra$i(^{\backslash }J1$$n^{2}$$\triangle$$\epsilon$$\Delta(c_{i,j})=\sum_{k=1}^{n}c_{i.k}\otimes c_{k,j}$,$\epsilon(c_{i,j})=\delta_{i,j}$

and $A_{k}(n, r)$ makes a sub-coalgebra. Then $S_{k}(n,r)\cong Hom_{k}(A_{k}(n, r), k)$ as k-algebras.
(3) The categories of finite dimensional $S_{k}(n, r)$ -modules and polynomial representations ofof homogeneous degree are equivalent, see [14]. Since the symmetric power $S^{d}E$ is apolynomial representation of $GL_{n}(k)$ of degree , it follows that for any sequence $\alpha=(a_{1}, a_{2}, \ldots)$with $a_{1}+a_{2}+\cdots=r$ , the tensor product $S^{\alpha}E=S^{a_{1}}E\otimes S^{a}2E8\cdots$ is a $S_{k}(n, r)$ -module. If$d\geq p>0$ , then $\{v^{p}|v\in E\}$ generates a $GL_{n}(k)$ -submodule of $S^{d}E$ . Let
denote thequotient module (called truncated symmetric power), which is still homogenous of degree . In
particular, the truncated tensor symmetric power
is a $S_{h}.(n, r)-lnodule$ .
$GL_{n}(k)$

$r$

$d$

$\Gamma^{d}E$$d$$\Gamma^{\alpha}E=\Gamma^{a_{1}}E\otimes\Gamma^{a}2E\otimes\cdots$(4) The categories of $S_{k}(n, r)$ -modules and $A_{k}(n, r)$ -comodules are canonically isomorphic.

Given a $S_{k}(n, r)$ -module, or equivalently an $A_{k}(n, r)$ -comodule $M$ , let : $Marrow M\otimes A_{k}(n, r)$ be
the structure map. Choose a basis
of $M$ . Thenwith $f_{j,i}\in A_{k}(n, r)$ .
The coefficient space $cf(M)$ is defined to be the k-span of all. It is easy to check that $cf(l1I)$
is a sub-coalgebra of $A_{k}(n.r)$ and is independent of the choice of bases.
$\delta$$\{m_{i}\}$$\delta(m_{i})=\sum m_{j}\otimes f_{j,i}$$f_{j,i}$We collect necessary properties of Schur algebras (q-Schur algebras) as below, see also [4, 14].
(A) There exists an involution (anti-automorphism) of k-algebras: $S_{k}(n, r)arrow S_{k}(n, r)$which preserves a complete set of orthogonal primitive idempotents. Call a duality later on.
$\omega$$\omega$(B) There is a bijection between the iso-classes of irreducible $S_{k}(n, r)$ -modules and

For each
cover of(C)$\lambda\in\Lambda^{+}(n, r)$$L(\lambda)$, let$L(\lambda)$

be the corresponding irreducible module and$P(\lambda)$$\Lambda^{+}(n,r)$.

the projective.

$\Lambda^{+}(n, r)$

is a poset with respect to the dominance ordering:$\lambda\geq\mu=\lambda_{1}+\cdots+\lambda_{\epsilon}\geq\mu_{1}+\cdots+\mu_{s}$,$\forall s\geq 1$

Moreover, $S_{k}(n, r)$ is quasi-hereditary in the sense of Cline, Parshall and Scott [2], see also Section
2.3. For each, there exists the Weyl moduleand the tilting module.
Moreover, eachadmits a filtration by Weyl modules and the multiplicity
isindependent of choice of filtrations and thus well-defined.
$\lambda\in\Lambda^{+}(n, r)$$T(\lambda)$$\triangle(\lambda)$$T(\lambda)$$[T(\lambda) :(D) (Schur-Weyl duality) We have the surjective algebra morphism\Delta(\mu)]$$k\Sigma_{r}arrow End_{S_{k}(n,r)}(E^{\copyright r})$.

If $n\geq r$ , then there exists an idempotent $e=e^{2}\in S_{k}(n, r)$ such that $E^{\otimes r}\cong S_{k}(n, r)e$ andEnd. Moreover,is a projective and injective faithful as a
$S_{k}(n, r)$ -module.
$k\Sigma_{r}\cong$$S_{A}.(n,r)(E^{\copyright r})\cong eS_{k}(n, r)e$$E^{\otimes r}\llcorner$

50: $S_{k}(n, r)-mod arrow k\Sigma_{r}$-mod be the functor which sends a (left) $S_{k}(n,r)$ -module $M$
to $f(M)=Hom_{S_{k}(n,r)}(E^{\otimes r}, M)$ . Then is exact when $n\geq r$ and in this case sends
to
to
and
sgn, where is the transpose of .
to(E) Let

$f$

$f$

$f$

$S_{\lambda},$$P(\lambda)$$Y^{\lambda}$$Y^{\lambda^{l}}\otimes$$T(\lambda)$$\lambda’$$\Delta(\lambda)$$\lambda$

Theorem 2.2. (Hemmer, Kleshchev and Nakano [16, 19]) If $p\geq 5$ and $n\geq r$ , then thefunctor in (E) induces isomorphisms$f$$Ext_{S_{k}(n,r)}^{i}$$(M. N)\cong Ext_{k\Sigma}^{i},$$(f(M), f(N))$ ,$0\leq i\leq p-3$

for any which has a Weyl module filtration and all M. In particular, the filtmtion multiplicitiesof dual Specht modules in Young modules are well-defined. In the quantum case, the quantum$N$

Weyl modules and the functor are defined similarly. Let be the quantum chamcteristic. If$n\geq r$ and $l\geq 4$ , theninduces isomorphism $Ext_{S_{q}(n.r)}^{1}(M, N)\cong Ext_{H_{q}(r)}^{1}(f(Af), f(N))$ for$N$
any which has a quantum Weyl module filtration and all $M$ .
$l$

$f$

$f$

2.2

Schur functor

Let be a finite dimensional algebra over an arbitrary field and denote by A-mod the categoryof finite dimensional left A-modules. Let $e=e^{2}$ be an idelnpotent in . Then $Ae$ is a projective
left A-module with the endomorphism algebra $End_{A}(Ae)\cong eAe$ . Consider the exact fumctor
$A$

$k$

$A$

$f$

:

$A-mod arrow eAe-mod$$M\mapsto Hom_{A}(Ae, M)\cong eM$

and call it a general Schur functor. In [5], a Grothendieck spectral sequence is established forgeneral Schur functors, namely for any $A_{-1}noduleM$ and eAe-mo dule $N$$E_{2}^{i,j}=Ext_{A}^{i}(M, Ext_{eAe}^{j}(eA, N))\Rightarrow Ext_{eAe}^{i+j}(eM, N)$

2.3

Standarly stratified and quasi-hereditary algebra

Quasi-hereditary algebras were introduced by Cline, Parshall aiid Scott to study highest weightcategories [2]. By definition. an algebra is said to be quasi-hereditary over a poset $X$ if there
is a bijection from $X$ to the iso-classes of simple A-modules and for each $x\in X$ , there is aquotient module
of $P(x)$ , called a standard module, satisfying$A$$\Delta(x)$(1) the kernel of the canonical morphism $\Delta(x)arrow L(x)$ is filtered by $L(y)$ with $y<x$ ;(2) the kernel of the canonical morphism $P(x)arrow\Delta(x)$ is filtered by$\Delta(z)$with $z>x$ .

of the
The costandard module
is defined to be the k-dual of the standard module$(A^{op},X)$(resp.
) we denote the full subcategory of
quasi-hereditary algebra. By
A-mod whose objects are filtered by standard modules (resp. costandard modules).
$\Delta_{A^{op}}(x)$$\nabla(x)$$\ovalbox{\tt\small REJECT}(\triangle)$$\ovalbox{\tt\small REJECT}(\nabla)$

Quasi-hereditary algebras have a ring-theoretical definition in terms of hereditary ideals, see[4, appendix]. The following contains some basic properties of these algebras.

51(1) For each $x\in X$ , there is a umique (up to isomorphism) indecomposable module $T(x)$incalled a tilting module. The direct sum $T;=\oplus_{x\in X}T(x)$ is called the$\ovalbox{\tt\small REJECT}(\triangle)\cap \mathscr{G}(\nabla)$.

chamcteristic tilting module of $(A, X)$ and add$(T)=\ovalbox{\tt\small REJECT}(\Delta)\cap\ovalbox{\tt\small REJECT}(\nabla)$.

(2) Let $Y$ be a cosatumted subset of $X$ (i.e., $x>y\in Y$ implies $x\in Y$ ). Let
be thecosaturated idempotent of $A$ corresponding to $Y$ . Then $I=AeA$ is a hereditary ideal in
$e$

$A$

and $(A/I, X\backslash Y),$ $(eAe, Y)$ are quasi-hereditary algebras. The standard modules of $A/I$ are
and the tilting modules of $A/I$ are $\{T(x) : x\in X\backslash Y\}$ . The standard$eAe$modules ofare $\{e\Delta(x) : x\in Y\}$ and the tilting modules of $eAe$ are $\{eT(x) : x\in Y\}$ .
$\{\Delta(x) :x\in X\backslash Y\}$

Standardly stratified \‘algebras were introduced to generalize quasi-hereditary algebras. We
remark that there are some variant definitions of standardly stratified algebra. , in Section 3, weshall adopt the definition due to Cline, Parshall and Scott. We will not give the definition here,
and the interested readers are referred to [3] or [7].
$s$Dominant dimension

2.4

Much of the theory of dominant dimension is due to Morita and Tachikawa, see [24, 25, 26, 28]
for more information and [1] for a treatment of some of the topics to be discussed below in terlnsof homological algebra.
be a finite dimensional algebra over an arbitrary field. The dominant dimension ofa left A-module $M$ , which we denote by dom. $\dim M$ is the maximal number (or ) such

Let

$A$

$t$

$\infty$that there exists an injective resolution $0arrow Marrow I_{0}arrow I_{1}arrow\cdotsarrow I_{t}arrow\cdots$ ofwithprojective for all $j<t$ (or ). Here we set dom. $\dim M=0$ if the injective hullof $M$ is
not projective or equivalently if $M$ is not a submodule of a module that is both projective andinjective. We write dom. $\dim A=$ dom. $\dim_{A}A$ . So, dom. $\dim A=0$ if and only if does nothave a faithful module that is both projective and injective. Algebras of dominaiit dimension atleast 1 are usually called QF-3 algebras (see [28]). If is QF-3, then there exists a unique (upto conjugate) idempotent $e=e^{2}\in A$ such that $Ae$ is projective, injective and minimal faithful.
To emphasize this idempotent, we shall denote a QF-3 algebra by $(A, e)$ .
$\Lambda I$$\infty$$I_{j}$$I_{0}$$A$$A$

If a QF-3 algebra $(A, e)$ has dominant dimension at least two, then the classical resultsof Morita, Tachikawa and others (see [26]) imply that there is a double centralizer property
between and its centralizer subalgebra $eAe:End_{A}(Ae)=eAe$ and $End_{eAe}(Ae)=A$ . In [21]it has been shown that classical Schur-Weyl duality between the Schur algebra $S_{k}(n, r)$ (for$n\geq r)$ and the group algebraof the symmetric group as well as Soergel ‘Struktursatz’
for the Bernstein-Gelfand-Gelfand category , providing a double centraliser property between
a block and a subalgebra of tlhe coinvariant algebra, both are special cases of this situation. In
particular, tliose Schur algebras and also blocks of have dominant dimension at least two.
$A$

$k\Sigma_{r}$$s$$\mathcal{O}$

$O$

If

is self-injective, then obviously dom. $\dim A=\infty$ . The Nakayama conjecture asserts thatthe converse also holds, i.e.,$A$dom. $\dim A=\infty\Leftrightarrow A$ is self-injective.

52
By use of the characterization of Schur algebras $S_{k}(p,p)$ in [27], we have dom. $\dim S(2,2)=2$
when $p=2$ and dom. $\dim S(3,3)=2$ , when $p=2$ . dom. $\dim S(3,3)=4$ when $p=3$ . From
the definition, it is generally very hard to compute the dominant dimension of a given algebra.
Nevertheless, tlie following characterization due to M\"uller is rather applicable.
Theorem 2.3. (M\"uller [23]) Assume that dom. $\dim A\geq 2$ and $Ae$ is a proj ective, injective
and faithful module. Let $M$ be an A-module. Then dom. $\dim M\geq n\geq 2$ if and only if $M\cong$$Honi_{eAe}(eA, eM)$ canonically and $Ext_{eAe}^{i}(eA, eM)=0$ for $1\leq i\leq n-2$ .

Connection between Schur functors and dominant dimension

3

Let $(A, e)$ be a QF-3 algebra. We shall consider the Schur functor afforded by the idempotent. As we have seen, this setting has included the most interesting cases like Schur-Weyl duality,
Soergel struktursatz and so on.
$f$

$e$

$s$

Note that by M\"ulller‘s characterization, we have $n(P_{A})=$ dom. $\dim A-2$ following thenotation from the introduction. This leads us to expect the connection between A-mod and$eAe$ -mod to be all the better the larger the $dominal$] dimension ofis. Indeed, we have
$t$

$A$

Theorem 3.1. Let $(A, e)$ be a QF-3 standanlly stratified algebm whose simples are indexed bya quasi-poset. If dom. $\dim A\geq P(X)+2+s$ for some $s\geq 1$ , then there is a full embedding, where $l(X)$ denotes the diameter of X. Furthermoreof categories: :$Ext_{A}^{i}(M, N)\cong Ext_{eAe}^{i}(eM, eN)$.
for $0\leq i\leq s$ for any $M\in A$ -mod and$(X. \theta)$$f$$\ovalbox{\tt\small REJECT}(\Delta)=\rangle eAe-mod$$N\in \mathscr{P}(\Delta)$

This theorem affirmatively hints at the importance of dominant dimension in the setting.
However, it is still weak to say anything deep. In the following, we restrict ourselves further toa proper class of algebras, which is rich enough to contain all interesting exainples mentioned
already, say (quantum) Schur algebras, block algebras of BGG category , etc.
$\mathcal{O}$

Definition 3.2. The class $d$ consists of finite dimensional k-algebms
the field and satisfy the following properties:$A$, which are split over$k$(1) $A$ is a quasi-hereditary algebm over a poset $X$ ,

(2)

$A$

has a dualityl.

(3) dom. $\dim A\geq 2$ ,Section 2.1 $(A)-(D)$ implies that $S_{k}(n, r)$ belongs to the class ,Of when $n\geq r$ . For simplicity,
we shall denote an algebra in the class .Of by a quadruple $(A, X,\omega, e)$ where is an idempotent
in such that $Ae$ is projective, injective and minimal faithful, $X$ is a poset indexing simple
A-modules and is the duality from the definition.
$A$

$A$

$\omega$lSee [9] for an exact definition of the duality here, see also Section 2.1 (A).

$e$

53
Theorem 3.3. Let $(A, X, \omega, e)$ be in the classandbe any A-module. Then the Schurhnctor induces canonical isomorphisms for any projective A-module $P$ and any$1|f$$K\in\ovalbox{\tt\small REJECT}(\Delta)$$Ext_{A}^{i}(M, P)\cong Ext_{eAe}^{i}(eM, eP)$,

$0\leq i\leq$,$0\leq i\leq dom.\dim T-2$ .

$Ext_{A}^{i}(M, K)\cong Ext_{eAe}^{i}(eM, eK)$

Furthermore, there are equalities$n(\mathcal{P}_{A})=$dom. $\dim A-2$

dom. $\dim A-2$ and$n(\ovalbox{\tt\small REJECT}(\triangle))=$dom. $\dim T-2$ .

From Theorem 3.3, we see that the Schur functor induces an equivalence fromthe full subcategoryof $eAe$-mod which consists of the eAe-rnodules filtered by$f$$\mathscr{G}(e\triangle)$$\mathscr{P}(\triangle)$to:$\{e\Delta(x)$provided dom. $\dim T\geq 2$ (or equivalently dom. $\dim A\geq 4$ by Theorem 3.4(4) below).
Moreover, this equivalence preserves Ext-groups up to degree dom. $\dim T-2$ . In this sense,$x\in X\}$

Theorem 3.3 largely extends Hemmer, Kleshchev and Nakano’s results for (quantized) Schuralgebras to all algebras in ,Of alld attributes Theorem 2.2 to the computation of doln. $\dim T$ .
The following theorem reduces further the computation of dom. $\dim T$ to that of dom. $\dim A$ . In
particular,
is independent of the choice of quasi-hereditary structures.
$n(\ovalbox{\tt\small REJECT}(\Delta))$

Theorem 3.4. Let $(A, X,\omega, e)$ be in the class .Of and$T$be the characteristic tilting module.

(1) The Ringel dual $R=End_{A}(T)^{\sigma p}$ belongs to $d$ ;(2) $R$ has characteristic tilting module $D(T)$ : dom. $\dim_{A}T\geq 1$ and doln. $\dim_{R}D(T)\geq 1$ ;(3) dom. $\dim_{R}D(T)=d_{0l}n.\dim_{A}T$ ;(4) dom. $\dim A=2$ dom. $\dim_{A}$ T. In parti cular, dom. $\dim A=$ dom. $\dim R$ .

Corollary 3.5. For any $x\in X$ , the standard module
has dominant dimension at least 1.
In particular. all standard A-modules are torsionless (i.e., submodules of projectives).
$\Delta(x)$This generalizes results of James and Donkin [4] on the torsionless property of (quantum)
Weyl modules, see also [16]. For blocks of category , it is well-known that all Verma modules
are submodules of the projective Verma module. The following theorem describes the behaviorof dominant dimension under truncation process.
$\mathcal{O}$

Theorem 3.6. Let $(A, X, \omega, e)$ be an indecomposable algebm in the classmastmal element in the poset $X$ .
(1) If$T(\lambda_{\max})=\triangle(\lambda_{m\infty})$(2) If$T(\lambda_{m})/\Delta(\lambda_{m\infty})$, then $A$ is simple and isomorphic to$k$$\ovalbox{\tt\small REJECT}$. Let$\lambda_{mR}$

be a

.

is tilting, then dom. $\dim A=2$ and dom. $\dim A/I=2$ or$\infty$.

(3) If dom. $\dim A=n\geq 4$ , then dom. $\dim_{A/I}A/I=n$ or $n-2$ .

There seems to be a close relation between global dimension and dominant dimension as thefollowing corollary indicates.

Corollary 3.7. Let be an algebra in the class .Of and $i>0$ . Then $Ext_{A}^{i}(D(A), A)\neq 0$ implies
dom. $\dim A-1\leq i\leq$ gl. $di_{l}nA;Ext_{A}^{i}(T, A)\neq 0$ implies dom. $\dim T-1\leq i\leq$ proj. $\dim T$ , where$D(A)$ is the k-dual
of .
$A$

$A$

54
Generalized symmetric algebras and dominant dimension

4

In order to develop further llew characterizations and applications of dominant dimension, we
enlarged our algebra class ,Of in Section 3 to introduoe generalized symmetric algebras. The
new class contains on the one side some quasi-hereditary algebras (finite global dimension),
like (quantum) Schur algebras, block algebras of BGG category , and on the other side, allsymmetric algebras (infinite global dimension), say Hecke algebras. The following theorem
provides both a definition and equivalent characterizations of generalized sylnmetric algebras$\mathcal{O}$[10].

Theorem 4.1. Let$A$

be a finite dimensional k-algebra. The following statements are equivalent:(1) dom. $\dim A\geq 2$ and $D(Ae)\cong eA$ as $(eAe, A)$ -bimodules, where $Ae$ is a basic faithfulprojective and injective module;(2) $Hom_{A}(D(A), A)\cong A$ as $(A, A)$ -bimodules;(3) $D(A)\otimes_{A}D(A)\cong D(A)$ as $(A, A)$ -bimodules;(4) $A$ is the endomorphism ring of a generator over a symmetmc algebra.

As an application, we obtained an refinement of the Tachikawa-Morita correspondence, i.e.,$A$finite dimensional$\{(\Lambda, M)|_{alga_{t>}enerat\circ rin\Lambda-mod}^{\Lambda finitedimensiona1symnet_{1}\cdot ic}ebra,M\sigma\}\{A|Hom_{A}(D(A), A)\cong A$$\}$

as $(A, A)$ -bimodules. Here denotes the duality over the ground field. We remark
which sendstothat generators over symmetric algebras are the same as cogenerators, therefore, no confusion willarise when talking about generators instead of generator-cogenerators for symmetric algebras.
Another feature of generalized symmetric algebras is that they admit a new characterization of
dominant dimension [10].
$(\Lambda, M)$$End_{\Lambda}(M)$$D$.

Proposition 4.2. Let be a generalized symmetric k-algebra. Then for any left A-module $M$
we have dom. $\dim M\geq n$ if and only if $Hom_{A}(D(A), M)\cong M$ and $Ext_{A}^{i}(D(A)$ , At $=0$ for$i=1,2,$$n-2$, where is an integer no less than 2.
$A$

$)$

$\ldots,$$n$

In [11], we construct a Hochschild cocomplex for every generalized symmetric algebras andstudy its exactness. As a byproduct, we obtained a relation between dominant dimension and
Hochschild cohomology group. Indeed, by Theorem 4.1(3), every generalized sylnmetric algebra
satisfies $D(A)\otimes_{A}D(A)\cong D(A)$ as $(A, A)$ -bimodules. Thus the composite$A$$D(A)\otimes_{k}D(A)arrow mD(A)\otimes_{A}D(A)\cong D(A)$defines an associative multiplication on $D(A)$ . We remark that $D(A)$ with this multiplication $m$
usually has no unit and we call prove that $(D(A), m)$ has a unit if and only if is symmetric.
Consider the bar complex of the algebra $(D(A), m)$ .
$A$$...arrow m3D(A)\otimes D(A)\otimes D(A)arrow 2D(A)m\otimes D(A)arrow m_{1}:=mD(A)arrow 0$

55
where
$D(A)$$m_{i}(f_{0} \otimes\cdots\otimes f_{i})=\sum_{j=0}^{i}(-1)^{j}f_{0}\otimes\cdots\otimes f_{j-1}\otimes m(f_{i}, f_{i+1})\otimes f_{i+2}\Theta\cdots\otimes f_{i}$for$f_{1},$$\ldots,$$f_{i}\in$. Dualizing the complex, we obtain a Hochschild cocomplex$0arrow Aarrow A\otimes_{k}Aarrow A\otimes_{k}$ $A$ $\Theta_{k}Aarrow\cdots$

We note that the cocomplex is not canonically determined byfor the choice of the isomorphism$D(A)\Phi_{A}D(A)\cong D(A)$ . However, different choices will result in isomorphic cocomplexes. Notealso that if $(D(A), m)$ has a unit, by the usual arguments, the cocomplex or its dual above isexact, whereas generally the cocomplex is not necessarily exact by the following theorem [11].
$A$

Theorem 4.3. Let
be a generalized symmetric algebra. Let
dom. $\dim A\geq n$ iff the sequence
$A$

$n$

be a natuml number. Thenis exact.

$0arrow Aarrow A^{\otimes 2}arrow\cdotsarrow A^{\otimes n}arrow A^{\otimes(n+1)}$

By definition, dom. $\dim A^{op}=$ dom. $\dim A$ . Regarding$A$

as an $(A, A)$ -bimodule, we have

Theorem 4.4. Let be a genemlized symmetric algebm. Then dom. $\dim A^{e}=$ dom. $\dim$dom. $\dim_{A^{e}}A=$ dom. $\dim A$ .
$A$

$A$

and

This theorem combined with Theorem 3.3 yields
Theorem 4.5. Let
be a genemlized symmetric algebra. Let $Ae$ be a faithful projective andinjective module. Then $HH^{i}(A)\cong$ HH’ $(eAe)$ for $0\leq i\leq$ dom. $\dim A-2$ .
$A$

5Computation of dominant dimension

With the techniques developed in Section 4 and’s theorem, we are able to compute thedomiliant dimension of a number of algebras. To be more precise, we have in [9]$M\ddot{u}1ler$

Theorem 5.1. Let be an infinite field of $chamcter\iota sticp>0$ . Let and be two natuml
numbers. If $n\geq r\geq p$ . then dom. $\dim S_{k}(n, r)=2(p-1)$ and $n(\mathscr{P}(\triangle))=p-3$ .
$k$

$n$

$r$

Note that if $p=0$ or $p>r$ , the Schur algebra $S_{k}(n, r)$ is semisimple and hence has
dominant dimension. We remark that there are now three different proofs available for thistheorem and a quantum analogue of the theorem also exists. Namely, let be the quantum
characteristic and assume that $n\geq r\geq\ell$ . Then dom. $\dim S_{q}(n, r)=2(P-1)$ . In [7], we
computed the dominant dimension of block algebras of BGG categoryof semisimple Liealgebras, i.e.,$\infty$$\ell$$\mathcal{O}$

Theorem 5.2. Let be a non-seimsimple block algebra of the category
algebra. Then dom. $\dim A=2$ .
$A$$\mathcal{O}$of a semisimple Lie

In [12], we made an attempt to compute the dominant dimension of the Schur algebra
for general $n,$ . Note that the tensor space
becolnes a projective, injective faithful$S_{k}(n, r)$ -module when $n\geq r$ , see Section 2.1 (D), but fails to be projective when $n<r$ . Even
worse, weknow a clear candidate of a projective, injective and faithful $S_{k}(n, r)$ -modulein this latter case. As a result, we are left almost no chalice to apply M\"uller characterization$S_{k}(n, r)$$r$$E^{\Gamma}\emptyset r$$don^{:}t$

$s$

56
directly. Our strategy below is to make an essential use of the theory of dominant dimension
newly developed in Section 4.
be a commutative ring with a unit. Let $M_{n}(R)$ be the algebraic monoid of
matricesover and let $A_{R}(n, r)$ be the space of homogenous polynomials of degree in -indeterminants. Clearly $A_{R}(n, r)$ is a free R-module of finite rank. Let $S_{R}(n, r)=Hom_{R}(A_{R}(n, r), R)$ . Then
we have by Section 2.1 (see also [14]) that, $A_{R}(n, r)\otimes_{S_{R}(n,r)}A_{R}(n,r)\cong A_{R}(n, r)\otimes_{\lambda t_{n}(R)}A_{R}(n, r)$
Let

$R$

$n\cross n$

$R$

$r$

$n^{2}$$q_{j}$

and $A_{k}(n, r)\cong A_{z}(n,r)\otimes_{Z}k.S_{k}(n,r)\cong S_{z}(n,r)\otimes_{z}k$ for any field .
$k$Lemma 5.3. Let be any infinite field. Then the Schur algebra $S_{k}(n, r)$ utth $n\geq r$ is generalized
symmetric. In particular, $A_{k}(n,r)\otimes_{S_{k}(n.r)}A_{k}(n, r)\cong A_{k}(n,r)$ as $S_{k}(n, r)$ -bimodules.
$k$With the aid of this lemma, we have that for all$n,$ $r$

Theorem 5.4. $A_{R}(n, r)\otimes_{M_{n}(R)}A_{R}(n, r)$ is a free R-module.
Note that $A_{R}(n, r)$ is spanned by the monomial

with$c_{\underline{\tau},\underline{j}}=q_{1},\cdots c$

Lemma 5.5. In $A_{R}(n, r)\otimes_{M_{n}(R)}A_{R}(n, r)$ , we have that$q_{\underline{j}}\overline{\otimes}c_{\underline{k},\underline{l}}\neq 0$

implies$\underline{i},\underline{j}\in I(n, r)$$\underline{i}\sim\underline{l}$.

.

Theorem 5.6. Let be a commutative ring with a unit. Let and be two natural numbers.
Then the map $\Theta=\Theta_{n,r.R}:A_{R}(n, r)\otimes_{M_{n}(R)}A_{R}(n, r)arrow A_{R}(n, r)$ defined by
$R$

$n$

$\sum$$\Theta(q_{-},\underline{j}\otimes c_{\underline{k},\underline{i}})=$$\sigma\in$Stabis a $S_{R}(n, r)$ -bimodule homomorphism. If $n\geq r$ , then$r$$c_{\underline{k}.\underline{j}\sigma}$$(\underline{i})$$\Theta_{n,r,R}$

is an isomorphism.

Combined with Lemma 5.5, to establish the morphism , we only need to check that it
is well-defined. We note thatis combinatorially defined, but fails to be either injective
or surjective for general and . Let be an infinite field of characteristic $p=2$ and $n=$$c_{11}e_{2}+$$r=3$ . Then the image ofis spanned by, which is definitely not. Indeed, as we shall see below, theimage of.
coincides with the Doty coalgebra$\Theta$$\Theta_{n,r,R}$

$n$

$2,$

$r$

$k$

$\{c_{11}^{2}c_{22}+c_{11}c_{12}c_{21},$ $c_{11}c_{12}c_{22}+c_{12}^{2}c_{21},$$\Theta_{2,3,k}$$c_{12}c_{21}c_{22},c_{11}c_{21}c_{22}+C_{12}e_{1}\}$$A_{k}(2_{!}3)$$\Theta_{n.r.k}$$D_{n,r,p}$

be a field of characterzstic $p>0$ . The Doty coalgebra $D_{k}(n, r)$ $:=D_{n,r,p}$
is defined to be the sum of the coefficient space of the truncated tensor symmetric power
and $a_{1}+a_{2}+\cdots=r$ , see Section 2.1.
for all sequences $a=(a_{1}, a_{2}, \ldots)$ with

Definition 5.7. Let$k$$\Gamma^{\alpha}E$$a_{i}\geq 0$

Doty coalgebras were introduced by Doty and Walker in order to study the decomposition
matrix of general linear groups [6]. Recently, Heaton made a systematic investigation on whena Doty coalgebra is quasi-hereditary (i.e., the dual algebra is quasi-hereditary) [15]. In [12], weproved the following relation between the map.
and the Doty coalgebra$D_{n,r,p}$$\Theta_{n,r,k}$

Proposition 5.8. Let and5.6. Then ${\rm Im}(\Theta_{n,r.k})=D_{n,r}$ .
$n$

$r$

be two natuml numbers and$\Theta_{n,r.k}$be the map from Theorem

Theorem 5.9. Let be an infinite field of charactertstic $p>0$ . Let and be two natural
numbers. $Let\ominus;=\Theta_{n,r.k}$ be the multiplication map on $A_{k}(n, r)$ from Theorem 5.6. Then thefollowing statements are equivalent
$k$

$n$

$r$

57
(1)

$\Theta$

is an isomorphism;(2) $D_{n,r}=A_{k}(n, r)$ ;(3) $r\leq n(p-1)$ .

We remark that (1) the proof of this theorem involves straightening bideterminants of boson
type following the line in [8]; (2) the theorem supplements Heaton’s result and colnbined with
Theorem 4.1 yields

Corollary 5.10. Let be an infinite field of charactenstic $p>0$ . Let and be ) natuml
numbers. Then $S_{k}(n, r)$ is genemlized symmetric and hence has dominant dimension at least 2
whenever $r\leq n(p-1)$ .
$k$

$n$

$r$

$tu$ $0$

With this corollary and Theorem 4.3 in Section 4, we are able to compute dom. diln $S_{k}(n, r)$using computer programme in case $r\leq n(p-1)$ . On the other hand, the condition in thiscorollary is only sufficient but not necessary as the following example illustrates, see also [22].
Let be a field of characteristic 2. Then the Schur algebra $S_{k}(2.7)$ is Morita equivalent to$S_{k}(2,2)\cross k\cross k$ . In particular dom. $\dim S_{k}(2,7)=$dom. $\dim S_{k}(2,2)=2$ .
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[11] M. Fang and S. Koenig, Hochschild cocomplex, Hochschild cohomology and dominant dimension, prepnnt.
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$Z$.

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