Schur–Weyl dualities for the rook monoid: an approach via Schur algebras

The rook monoid, also known as the symmetric inverse monoid, is the archetypal structure when it comes to extend the principle of symmetry. In this paper, we establish a Schur–Weyl duality between this monoid and an extension of the classical Schur algebra, which we name the extended Schur algebra. We also explain how this relates to Solomon’s Schur–Weyl duality between the rook monoid and the general linear group and mention some advantages of our approach


Introduction
Throughout this article, F is a field of characteristic zero unless explicitly specified and V is a d-dimensional vector space over F. The symmetric group S n acts on the tensor space ⊗ n V by place permutations.By fixing a basis of V , GL(V ) can be identified with the general linear group of all d × d non-singular matrices with entries in F, herein denoted G d .If V is the natural module for the group algebra FG d , then G d acts diagonally on ⊗ n V .This action commutes with that of S n on ⊗ n V by place permutations.In case F = C, I. Schur [32] established that each action generates the full centralizer of the other on End F (⊗ n V ), a result which was made popular by H.
Weyl [39].This seminal example of a double centralizer phenomenon, now known as the classical Schur-Weyl duality, provides a deep insight on the interactions between the representation theories of G d and S n .
Results of R. M. Thrall [37], C. de Concini and C. Procesi [5], R. Carter and G. Lusztig [4] and J. A. Green [11] show that classical Schur-Weyl duality remains true if F is an infinite field of any characteristic.More recently, classical Schur-Weyl duality was extended to sufficiently large finite fields by D. Benson and S. Doty [1].
These results can be better understood in the context of Schur algebras [11].Implicit in zero characteristic in Schur's Ph.D thesis [31], Schur algebras were defined over arbitrary infinite fields by J. A. Green in a seminal monograph [11].The Schur algebra S F (d, n) can be identified with the centralizer algebra End FSn (⊗ n V ) of FS n on ⊗ n V with respect to place permutations and the family {S F (d, r)} r≥0 completely determines the polynomial representations of FG d .Moreover, S F (d, n) is an important example of a cellular algebra [10].Thus, the classical Schur-Weyl duality can be stated in terms of these finite-dimensional algebras in a very general setting.
There are numerous other examples of "Schur-Weyl dualities".For instance, in characteristic zero, the centralizer algebras associated with the diagonal action of subgroups of G d such as the orthogonal group O d and the symmetric group S d on ⊗ n V are, respectively, the Brauer algebra [3] and the partition algebra [18,21,22,24] (see also [16]).As before, the translation of these results in the language of Schur algebras and their generalizations has widely expanded our knowledge of the properties of these algebras in the modular case (see, among many others, [2,7,8]).
In 2002, L. Solomon [34] established a Schur-Weyl duality between G d and an important finite inverse monoid.Inverse monoids were introduced in [38] as a natural generalization of groups to deal with aspects of symmetry which the latter couldn't capture (see [20] for further details on this viewpoint).The archetypal example of such a structure is the symmetric inverse monoid, also called the rook monoid [33].
For our purposes, the rook monoid R n is the set of all bijective partial maps from n = {1, . . ., n} to itself under the usual composition of partial functions.It contains S n and it is isomorphic to the monoid under matrix multiplication of all n×n matrices with at most one entry equal to 1 F in each row and in each column and zeros elsewhere.It plays the same rôle for inverse monoids that S n does for groups and thus is the archetypal structure when it comes to extend the principle of symmetry.
In his influential article [34], L. Solomon proved that R n acts on tensors by "place permutations".More precisely, he showed that, if F has characteristic zero and d ≥ n, FR n acts as the centralizer algebra for the action of G d on ⊗ n U , where U = V ⊕ U 0 is the direct sum of the natural d-dimensional module V and the trivial module U 0 .
Since its publication, this result proved to be a special case of an important Schur-Weyl duality on tensor spaces for the Hecke algebra analog for R n , known as the q-rook monoid (see [14,29,35] and references therein).It also influenced other authors into establishing Schur-Weyl dualities between R n and other finite inverse semigroups [19].Moreover, it led to the investigation of a number of interesting algebras.For instance, the centralizer algebras associated with the restriction of the action of G d on ⊗ n U to subgroups such as the orthogonal subgroup O d and the symmetric S d are, respectively, the rook Brauer algebra [15,23] and the rook partition algebra [12].
The main purpose of this article is to show that Solomon's Schur-Weyl duality for R n and G d can be stated in terms of an extension of the classical Schur algebra.
We achieve this by defining an F-algebra S F (d, n) which we call the extended Schur algebra and which satisfies We then prove that S F (d, n) determines the homogeneous polynomial representations of FG d of degree at most n, a result that holds for arbitrary infinite fields.Finally, we establish a Schur-Weyl duality between S F (d, n) and R n on ⊗ n U , when d ≥ n and F has zero characteristic.To show that our viewpoint provides a deeper insight on the representation theory of R n and its interactions to those of general linear and symmetric groups, we also mention some applications of our approach.
This paper is organized as follows.Section 2 begins with a brief overview on (split) semisimple algebras, double centralizer theory and classical Schur-Weyl duality.This is followed by a description of structural aspects of the classical Schur algebra S F (d, n) and an outline of the representation theory of the rook monoid R n .
In Section 3, we view G d ⊆ G d+1 under a natural embedding and we explain how the restriction of the diagonal action of G d+1 on ⊗ n U to G d gives rise to the extended Schur algebra S F (d, n).After describing this algebra's structure, we prove that the module category for S F (d, n) is equivalent to the category of homogeneous polynomial G d -modules of degree at most n.Finally, we establish a Schur-Weyl duality on ⊗ n U between S F (d, n) and FR n .We end by explaining how this result relates to Solomon's Schur-Weyl duality [34] and mentioning some consequences of our approach.
We should note that some of the techniques used herein apply to infinite fields of any characteristic.The fact that our main result relies on the semisimplicity of the monoid algebra of R n has made us decide to work in characteristic zero.However, since we hope to treat the modular case in the near future, we have pointed out all the results in this article that remain true for arbitrary infinite fields.

Double centralizer theory and classical Schur-Weyl duality
Henceforth, the term "module" refers to a finite-dimensional left module unless explicitly stated otherwise and F is a field of characteristic zero.Let A be a finite-dimensional split semisimple algebra over F. By classical Artin-Wedderburn theory [6,Theorem 3.34], this means that there is an isomorphism of F-algebras for some finite index set Λ and positive integers d λ .For each λ ∈ Λ, there is, up to isomorphism, one simple A-module S λ and {S λ : λ ∈ Λ} is a complete set of representatives of the isomorphism classes of simple modules of A.
If M is a finite-dimensional A-module, its decomposition into simple A-modules is given by where m λ is a non-negative integer called the multiplicity of λ in M .We say that λ ∈ Λ appears in M if M contains a submodule isomorphic to S λ (that is, if m λ > 0).Let ρ : A → End F (M ) be the representation corresponding to the A-module M with decomposition given by Expression (1).The centralizer algebra of A on M is the finite-dimensional F-algebra The action of End A (M ) on M defined by φ • x = φ(x), with φ ∈ End A (M ) and x ∈ M , turns M into an End A (M )-module.Since the actions of A and End A (M ) on M commute, ρ induces a homomorphism of F-algebras ρ : A → End End A(M ) (M ).In case M is a right A-module, the corresponding homomorphism is obtained via the representation ρ : The next theorem summarizes the basic dual relationship between the F-algebras A and End A (M ) and can be found (with a different formulation) in [6, Section 3.D].
Theorem 1 (Double Centralizer Theorem).Let A be a finite-dimensional split semisimple F-algebra and let {S λ : λ ∈ Λ} be a complete set of representatives of the isomorphism classes of simple A-modules with dim induces an isomorphism of F-algebras i.e., the actions of A and End A (M ) on M generate the full centralizer of the other in End F (M ).
Schur-Weyl duality is a cornerstone of representation theory that amounts to two double centralizer results which involve the symmetric and general linear groups.
Since F has characteristic zero, it is known that the group algebra of the symmetric group FS n is a split semisimple algebra of dimension n !(see, for example, [17, Theorem 5.9.]).On the other hand, the isomorphism classes of simple modules for FS n are indexed by partitions of n.
Let V be an F-space of dimension d.For reasons that will become apparent later, we view ⊗ n V as a right FS n -module on which S n acts by place permutations as for all v 1 , . . ., v n ∈ V and σ ∈ S n .Let G d be the general linear group of all d × d invertible matrices with entries in F. If V has as F-basis {e 1 , . . ., e d }, we consider the natural left action of G d on V , defined on basis elements by where, for all for all v 1 , . . ., v n ∈ V and g ∈ G d .Hence, ⊗ n V is both a left FG d -module and a right FS n -module via place permutations.It is easily seen that these actions commute.In case F = C, the following theorem goes back to Schur's famous 1927 article [32].As mentioned in the introduction, the result holds over infinite fields of any characteristic.
Theorem 2. Let V be a d-dimensional vector space over F and regard ⊗ n V both as the left diagonal FG d -module and the right FS n -module by place permutations.
then ρ is injective and thus it induces an isomorphism of F-algebras

The classical Schur algebra
Throughout this paper, we shall adopt Green's viewpoint [11] on the polynomial representations of G d and hence work with Schur algebras (see also [25]).All of the results presented in this section are valid for infinite fields of arbitrary characteristic.
For all 1 ≤ i, j ≤ d, let c i,j : G d → F be the previously defined (i, j)-th coordinate function.The polynomial algebra A(d) ≡ F[c i,j : 1 ≤ i, j ≤ d] has the structure of a bialgebra with comultiplication and counit given by For each non-negative integer n, let A(d, n) be the F-subspace of homogeneous polynomials of degree n.The bialgebra A(d) has a natural grading as ) is an associative algebra over F which we denote S F (d, n) and call the (classical) Schur algebra.
We shall need some notation.If n ≥ 1, recall that n = {1 , . . ., n}.We write Γ n (d) for the set of all maps α : n → d, identifying each α with the n-tuple (α(1), . . ., α(n)) or, equivalently, (α 1 , . . ., α n ).The symmetric group S n acts on the right on Γ n (d) by the rule ασ The space A(d, n) has as F-basis the set of all monomials of degree n in the d 2 variables c i,j .Each such monomial can be written as c α,β = c α1,β1 . . .c αn,βn for some α, β ∈ Γ n (d) and there is an "equality" rule [11, Equation (2.1b)] for all α, β, γ, ν ∈ Γ n (d).Let Ω n be an arbitrary set of representatives of the We write {ξ α,β : (α, The algebra structure on S F (d, n) is the dual of the coalgebra structure on A(d, n).This implies the following rule for the multiplication in The importance of Schur algebras stems from the fact that they determine the finite-dimensional polynomial representations of FG d .We say that a representation ρ : FG d → End F (W ) on an F-space W is polynomial if its coefficient functions lie in A(d) and homogeneous of degree n if its coefficient functions lie in A(d, n).Following Schur's arguments for F = C, J. A. Green proved that every polynomial representation of FG d is a direct sum of homogeneous ones [11,Theorem 2.2c] and that the category of S F (d, n)-modules is equivalent to the category of homogeneous polynomial modules of FG d of degree n (see [11, pp. 23 − 25]).
For our purposes, we focus on the FG d -module ⊗ n V with action given by Expression (4).If {e 1 , . . ., e d } is an F-basis of V , then {e ⊗ α = e α(1) ⊗ . . .⊗ e α(n) : α ∈ Γ n (d)} is an F-basis of ⊗ n V .Hence, the left diagonal FG d -action can be expressed as Since the c α,β all lie in A(d, n), this amounts to saying that It is easily seen that the previous S F (d, n)-action commutes with the right action of S n on ⊗ n V given by place permutations.In truth, the Schur-Weyl duality between G d and S n on ⊗ n V can be stated in terms of S F (d, n).The proof of the first isomorphism exhibited in the following theorem can be found in [11,Theorem (2.6c)].The other isomorphism follows from [4, p. 209, Lemma] and [11,Section 2.4].
Theorem 3. Let V be a d-dimensional vector space over F. Regard ⊗ n V both as a left S F (d, n)-module with action given by Equality (8) and a right FS n -module with action given by Equality (2).If d ≥ n, then each action generates the full centralizer of the other on End F (⊗ n V ) and hence, as F-algebras,

Representations of the rook monoid
The representation theory of finite inverse semigroups was established in the fifties by W. D. Munn [26][27][28] and I. S. Ponizovski ȋ [30].For the special case of the rook monoid, their results were furthered and deepened in zero characteristic by L. Solomon [34].
Recall that n = {1, . . ., n}.Our convention for the multiplication in R n is that the composition στ of the elements σ, τ ∈ R n is defined by first applying τ and then σ.If σ ∈ R n , we write D(σ) ⊆ n for the domain of σ and R(σ) ⊆ n for its range.We agree that R n contains a map ǫ ∅ with empty domain and range which behaves as a zero element in R n .With this convention, it is easy to see that we define the rank of σ, denoted rk(σ), as the size of its domain.We adopt the convention that the only element in R n of rank zero is ǫ ∅ and that S 0 = {ǫ ∅ } is a group with a single element.Note that any σ ∈ R n of rank n is a permutation of n and thus S n ⊆ R n .A minute's thought reveals that S r ⊆ R n , for r = 0, 1, . . ., n.
The set of idempotents of R n is the commutative submonoid of all the partial identities ǫ X : X → X, with X ⊆ n.For each X ⊆ n, ǫ X R n ǫ X is a monoid with identity ǫ X , whose group of units G X is called the maximal subgroup of R n at ǫ X .
Two idempotents ǫ X and ǫ Y are said to be isomorphic if G X ∼ = G Y as groups.If 0 ≤ r ≤ n and X ⊆ n is a set of size r, it is not hard to verify that Thus, each maximal subgroup of R n can be identified with some symmetric group S r .
In order to classify the isomorphisms classes of simple modules of FR n , we shall need some notation.If σ ∈ R n , we define the inverse of σ, denoted σ − , as the only element of R n which satisfies D(σ − ) = R(σ) and σ − σ = ǫ D(σ) .
If 0 ≤ r ≤ n and X ⊆ n is a set of size r, we denote by ι X the unique orderpreserving bijection between r and X (identifying 0 with the empty set).Note that any σ ∈ R n of rank r with D(σ) = X and R(σ) = Y can be mapped to S r via In particular, p(ǫ X ) = ǫ r ∈ S r , for all X ⊆ n of size r.
For our purposes, we also need to introduce special algebras.If 1 ≤ r ≤ n, let M ( n r ) (FS r ) be the F-algebra of all matrices with rows and columns indexed by subsets I, J of n of size r and entries in FS r .If r = 0, we identify M ( n r ) (FS r ) with F. Set If 1 ≤ r ≤ n and I, J ⊆ n are such that |I| = |J| = r, let E I,J be the standard matrix with ǫ r ∈ S r in position (I, J) and zeros elsewhere.If r = 0, set 1 The following result is essentially due to W. D. Munn [26,28] and I. S. Ponizovski ȋ [30] although it can be found explicitly in [34,Lemma 2.17] and [36,Theorem 4.4].
Theorem 4. Let F be a field of characteristic zero.There is an isomorphism of F-algebras φ : FR n → R n given by with inverse given on basis elements σ ∈ S r and E I,J with |I| = |J| = r by In particular, FR n is a finite-dimensional (split) semisimple algebra over F.
As a consequence of the previous theorem, the isomorphisms classes of simple modules of R n are in one-to-one correspondence with those of its maximal subgroups.In order to highlight the constructive aspect of Theorem 4, we express this result in terms of matrix representations of FR n .
Let µ ⊢ r with 0 ≤ r ≤ n and let ρ µ : FS r → M k (F) be an irreducible matrix representation of FS r .If r = 0, we agree that there is an empty partition µ = (0) and the corresponding irreducible representation of FS 0 is given by ρ It follows from Theorem 4 that ρ µ induces a matrix representation of FR n , denoted by ρ * µ : FR n → M k ( n r ) (F), and given by for all σ ∈ R n .The previous expression is due to L. Solomon (see Equation 2.26 in [34]).In the setting of finite inverse semigroups, W. D. Munn [28] showed that these representations determine the isomorphism type of FR n .
Theorem 5 (Munn).Let F be a field of characteristic zero.The set where g ∈ G d .All the results exposed in sections 3.1 and 3.2 remain valid if F is replaced by an infinite field of any characteristic.

A natural restriction
Let U = V ⊕U 0 be an F-space of dimension d+1, where V and U 0 are seen as subspaces of U of respective dimensions d and 1.As before, {e 1 , . . ., e d } is an arbitrary but fixed basis of V and U 0 = Fe ∞ for some vector e ∞ ∈ U which turns {e 1 , . . ., e d , e ∞ } into an F-basis of U .We assume the linear ordering 1 < 2 < . . .< d < ∞.
It follows from Section 2 that ⊗ n U is a finite-dimensional homogeneous polynomial FG d+1 -module of degree n via the diagonal action of G d+1 on ⊗ n U (see Equation ( 4) with d replaced by d + 1).We study this action's restriction to G d .
To do so, we need some notation.Let 1 ≤ r ≤ n and let X = {x 1 < . . .< x r } ⊆ n be a set of size r.We write Γ X (d) for the set of all maps α : X → d identifying α ∈ Γ X (d) with the r-tuple (α(x 1 ), . . ., α(x r )) ∈ d r .We also agree that Γ ∅ (d) has a single element which is identified with the empty set. If ) is as before a monomial of degree r in the d 2 variables c i,j .If ι X is the unique order-preserving bijection between r and X (see Section 2.3), the same monomial can be written as c α,β = c αιX ,βιX = c α(ιX (1)),β(ιX (1)) . . .c α(ιX (r)),β(ιX (r)) = c γ,ν , where γ = αι X , ν = βι X ∈ Γ r (d).This notation is particularly useful to represent the F-basis of ⊗ n U induced by {e 1 , . . ., e d , e ∞ }.If X ⊆ n and α ∈ Γ X (d), the decomposable tensor e ⊗ α ∈ ⊗ n U is defined as e ⊗ α = e α(1) ⊗ . . .⊗ e α(n) , where the map α : n → d ∪ {∞} is given by It is clear that ⊗ n U has as F-basis the set {e ⊗ α : α ∈ Γ X (d), X ⊆ n}.For instance, if d = 6, n = 5, X = {1, 3, 5} and α = (6, 2, 2) ∈ Γ {1,3,5} (6), then the corresponding basis element of ⊗ 5 U is given by Proposition 6.Let U be a (d + 1)-dimensional vector space over F. The restriction to G d of the left diagonal action of G d+1 on ⊗ n U is given by We turn to the action of g on basis elements of the diagonal FG d+1 -module , where β : n → d ∪ {∞} is defined as above.It follows that g • e β(i) = e ∞ , for all i / ∈ X and also that a α e ⊗ α , for some a α ∈ F. It is now easy to see that where c α,β ∈ A(d, r), for all α ∈ Γ X (d).

The extended Schur algebra
The coefficient space produced by the action of G d on ⊗ n U described in Proposition 6 suggests that ⊗ n U can be seen as a representation of a special Schur algebra.Let A n (d) = < c α,β : (α, β) ∈ Γ X (d) × Γ X (d), X ⊆ n > be the F-space spanned by all the monomials c α,β of degree at most n in the variables c i,j .A moment's thought and Equation (12) reveal that A n (d) is the direct sum of the first n + 1 homogeneous F-spaces A(d, r) of the graded bialgebra A(d).
It follows that A n (d) has as F-basis the set of all distinct monomials of degree r with 0 ≤ r ≤ n.By Equation ( 5), we index this basis with the set Ω 0 ∪ Ω 1 ∪ . . .∪ Ω n , where Ω 0 = Γ ∅ (d) × Γ ∅ (d) and each Ω r , with 1 ≤ r ≤ n, is a set of representatives of the S r -orbits of Γ r (d) × Γ r (d).
As mentioned previously, each A(d, r) is a subcoalgebra of A(d).Thus, A n (d) enherits a coalgebra structure with comultiplication and counit given, respectively, by for all α, β ∈ Γ X (d) and X ⊆ n.By a standard fact, it follows that is an associative F-algebra of finite dimension.Combined with Equation ( 6), this implies that dim for all γ, ν ∈ Γ k (d) and 0 ≤ k ≤ n.As with the classical case (see Equation ( 7)), if 0 ≤ k ≤ n and κ, ς ∈ Γ k (d), we also have an equality rule to take into account, namely, For now, it is enough to work with the index set n r=0 Γ r (d), since it follows from equations ( 12) and ( 15) that, for α with γ = αι X , ν = βι X ∈ Γ r (d) and ι X : r → X defined as before.
The multiplication in S F (d, n) follows from the coalgebra structure on A n (d) and hence from Equation (14).Thus, if ξ, η ∈ S F (d, n), the product ξη is defined on any monomial c α,β , with α, β ∈ Γ X (d) and X ⊆ n, by The unit element ǫ r , where ǫ r is the identity of S F (d, r).
We now turn to the tensor space ⊗ n U and show that it can be given the structure of a left S F (d, n)-module.We have however a stronger statement.
Proposition 7. The category of finite-dimensional FG d -modules whose coefficient 17) that e g e g ′ = e gg ′ and e I d = ǫ ∈ S F (d, n).Hence, the map e : g → e g can be linearly extended to an F-algebra homomorphism e : FG d → S F (d, n).
We assume that any map f : G d → F is identified with its unique linear extension f : FG d → F. Under this assumption, if k ∈ FG d , then e k : A n (d) → F is given by e k (c) = c(k), for all c ∈ A n (d).The arguments in [11, Proposition 2.4b, (i)] apply mutatis mutandis to e : FG d → A n (d) * and hence e is surjective.
We now show that f : FG d → F belongs to A n (d) if and only if f (k) = 0, for all k ∈ ker e.If f ∈ A n (d) and k ∈ ker e, then e k = 0 and e k (f ) = f (k) = 0. Conversely, let f : FG d → F be such that f (k) = 0, for all k ∈ ker e.Since e is surjective, for all ξ ∈ S F (d, n), there is some k ∈ FG d such that ξ = e k .Hence, we define y ∈ S F (d, n) * by y(ξ) = y(e k ) = f (k), for all k ∈ FG d .The condition that f (k) = 0, for all k ∈ ker e ensures that y is a well-defined element of Let V be a finite-dimensional left The proof is complete and we can now identify both categories by the simple rule: , for all k ∈ FG d and v ∈ V , where V is an object of either categories.
It follows from the proof of this result that the left action of We end this section with an important fact which follows from the semisimplicity of the classical Schur algebras.

Schur-Weyl duality between the rook monoid and the extended Schur algebra
In what follows, we describe the centralizer algebra End S F (d,n) (⊗ n U ) of S F (d, n) on the left module ⊗ n U on which S F (d, n) acts according to Equation (18).Throughout this section, F is a field of characteristic zero.
For each X ⊆ n, we denote by W X the F-subspace of ⊗ n U spanned by all the decomposable tensors e ⊗ α , with α ∈ Γ X (d).Since {e ⊗ α : α ∈ Γ X (d), X ⊆ n} is an F-basis of ⊗ n U , we have the following direct sum decomposition It follows from Equation ( 18) that the left action of an arbitrary ξ ∈ S F (d, n) on e ⊗ α , with α ∈ Γ X (d), is such that ξe ⊗ α ∈ W X .Hence, W X is an S F (d, n)-submodule of ⊗ n U and ( 19) is a decomposition of ⊗ n U as a direct sum of left S F (d, n)-submodules.
The next result shows how End S F (d,n) (⊗ n U ) decomposes into its building blocks.
(a) If 0 ≤ r ≤ n, the tensor space ⊗ r V is a left S F (d, n)-module for which there is an isomorphism of F-algebras , for all α ∈ Γ X (d), and extended linearly to W X , is easily seen to be a left S F (d, n)-module isomorphism.(c) This follows from (a) and (b) and the direct sum decomposition described in Equality (19).
It is worth noting that the previous lemma remains valid for arbitrary infinite fields.Recall that the algebra of matrices has an F-basis given by Equality (10) and that R n is isomorphic to FR n (Theorem 4).
Lemma 2. Let U be an F-space of dimension d + 1.Let α ∈ Γ X (d) for some X ⊆ n and let e ⊗ α be the corresponding basis element of ⊗ n U .If σ ∈ S r and I, J ⊆ n are such that Then Equation (20) gives ⊗ n U a right R n -module structure for which the action of R n commutes with that of S F (d, n) on ⊗ n U .
Proof.Let X ⊆ n and α ∈ Γ X (d).By the multiplication rules in R n , it suffices to check that (20) defines an action for basis elements of R n in the same block M ( n r ) (FS r ).The case r = 0 is trivial.If r ≥ 1, let σ, τ ∈ S r and I, J, K, L ⊆ n be sets of size r.Then We also have that e an equality obtained by reindexation (α = βσ −1 ⇔ β = ασ and σ(Y ) = X, for all X ⊆ n).This implies that (u 1 ⊗ . . .⊗ u n ) • σ = u σ(1) ⊗ . . .⊗ u σ(n) .
Thus, the restriction of the right R n -action on ⊗ n U given by Expression (21) to S n is the usual right action of S n on ⊗ n U by place permutations.
Although it is a lengthy computation, it is possible to show that the R n -action on ⊗ n U by "place permutations" defined by L. Solomon in [34,Equation (5.5)] turns into Expression (21) for basis elements of ⊗ n U .The reader should be aware that Solomon makes use of a different convention than ours when it comes to compose in R n .This means that Theorem 10 is a reformulation in the setting of Schur algebras of Solomon's important Schur-Weyl duality between G d and R n on ⊗ n U [34, Theorem 5.10].
It is worth noting that our approach opens up new possibilities for better understanding the extent of the interactions between the representation theories of rook monoids, general linear groups, symmetric groups and (extended) Schur algebras.
On one hand, our starting point was to view G d as a subgroup of G d+1 .This implies that a simple FG d+1 -module is also an FG d -module.Hence, it makes sense considering its decomposition into simple FG d -modules.In the language of Schur algebras, this amounts to decomposing a simple S F (d + 1, n)-module into simple S F (d, n)-modules and determining the corresponding multiplicities.This procedure is known as a branching rule for S F (d, n) ⊆ S F (d + 1, n) (see [13] and [9,Chapter 8]).Since both S F (d, n) and S F (d + 1, n) are finite-dimensional semisimple F-algebras, the branching rule for S F (d, n) ⊆ S F (d+1, n) is the same as that for End S F (d+1,n) (⊗ n U ) ⊆ End S F (d,n) (⊗ n U ) [13,Theorem 1.7.].By Theorem 10, this means that it is possible to derive concise proofs of combinatorial formulas for multiplicities associated with the restriction to S n ⊆ R n of irreducible characters of R n .In the near future, we hope to publish these proofs and recover in an economical way some of the results in [34,Section 3].
Another upshot of our approach is that we may use the tools associated with Schur algebras to give a construction of the irreducible modules of the rook monoid realized on tensors which is analogous to that of the dual Specht modules for the symmetric group.Indeed, we may apply Green's techniques [11,Chapter 6] and define an idempotent ζ ∈ S F (d, n) which satisfies the algebra isomorphism ζS F (d, n)ζ ∼ = FR n .The idempotent ζ induces a functor between the module categories for S F (d, n) and FR n and we can make use of this functor to build a complete set of simple modules for FR n from the Carter-Lusztig S F (d, n)-modules [4] realized on ⊗ n U , obtaining an analog for R n of the dual Specht modules for S n [11].We also expect to exhibit this construction in the near future.
Finally, in this article, we have laid the foundations for studying the modular representations of the rook monoid on tensors.Although Theorem 4 was stated in characteristic zero and our Schur-Weyl duality between R n and S F (d, n) on ⊗ n U relies heavily on this result, it is possible to show that a Schur-Weyl duality between (a subalgebra of) FR n and S F (d, n) on a tensor space can be established for infinite fields.

is a complete set
of inequivalent irreducible matrix representations of FR n .Thus, the isomorphism classes of simple modules of FR n are indexed by the set of partitions {µ ⊢ r : 0 ≤ r ≤ n} 3 Representing the rook monoid and the extended Schur algebra on tensors Throughout this section, F is a field of characteristic zero, d and n are positive integers such that d ≥ n and G d is viewed as the subgroup of G d+1 of all matrices of the form

)
Proof.Let g ∈ G d .Under the identification G d ≤ G d+1 , the natural action of g on basis elements of U becomes g • e j = d i=1 c i,j (g)e i + 0 .e∞ , for j = 1, . . ., d , and g • e ∞ = e ∞ .

Definition 1 .
Let d and n be positive integers.The extended Schur algebra for d and n over the field F, denoted S F (d, n), is the associative F-algebra of finite dimension given by the linear dualS F (d, n) = A n (d) * = Hom F (A n (d); F).Since A n (d) =n r=0 A(d, r), it follows that S F (d, n) can be regarded as the F-algebra n r=0 S F (d, r), where each S F (d, r) is a classical Schur algebra.Indeed, if ξ ∈ S F (d, r), we identify ξ with an element of S F (d, n) = n r=0 A(d, r) * by making it zero on all monomials whose degree is different from r.Under this identification, n r=0 {ξ α,β : (α, β) ∈ Ω r } is the F-basis of S F (d, n) which is dual to the F-basis of A n (d) given by n r=0 {c α,β : (α, β) ∈ Ω r }.
FG d -module with basis {v b : b ∈ B} and associated action g •v b = a∈B α a,b (g)v a , for all b ∈ B and g ∈ G d .Suppose that α a,b ∈ A n (d), for all a, b ∈ B. Then α a,b (k) = 0, for all k ∈ ker e and all a, b ∈ B. The actione g • v b = a∈B e g (α a,b )v a , for all g ∈ G d , b ∈ B, turns V into a left S F (d, n)-module.Indeed, for all ξ ∈ S F (d, n), there is k ∈ FG d such that ξ = e k .If ξ = e k = e k ′ for k ′ ∈ FG d , then k − k ′ ∈ker e and α a,b (k − k ′ ) = 0 for all a, b ∈ B. Hence, e k (α a,b ) = e k ′ (α a,b ) for all a, b ∈ B and the S F (d, n)-action is well defined.Conversely, if V is a left S F (d, n)-module with basis {v b : b ∈ B} and associated action ξ • v b = a∈B ξ(α a,b )v a , for all b ∈ B and all ξ ∈ S F (d, n), then V can be viewed as an FG d -module with action given by g • v b = e g • v b = a∈B α a,b (g), for all g ∈ G d and all b ∈ B, where e g = ξ ∈ S F (d, n) and α a,b (g) = ξ(α a,b ).Once again, the previous properties show that α a,b ∈ A n (d) and this action is well-defined.

Proposition 8 .
The extended Schur algebra S F (d, n) is a semisimple algebra over F. Proof.As referred previously, we may regard S F (d, n) as n r=0 S F (d, r), where S F (d, r) is the classical Schur algebra.A proof of the semisimplicity of S F (d, r) can be found in [11, Corollary (2.6e)] and hence the semisimplicity of S F (d, n) follows.
Proof.(a) If r = 0, we agree that ⊗ 0 V = F with trivial left S F (d, n)-action and right FS 0 -action.If r ≥ 1, we view ⊗ r V both as a left S F (d, r)-module (via Equation (8)) and a right FS r -module with respect to place permutations.The S F (d, r)-action on ⊗ r V is easily extended to an S F (d, n)-action by defining ξw = 0, for all w ∈ ⊗ r V and all ξ ∈ S F (d, k) with k = r.It follows that End S F (d,n) (⊗ r V ) ≡ End S F (d,r) (⊗ r V ) and, by Theorem 3, End S F (d,n)