Immanant varieties

We introduce immanant varieties, associated to simple characters of a finite group. They include well-studied classes of varieties, as Segre embeddings, Grassmannians and certain other classes of Chow varieties. For a one-dimensional character $\chi$, we define $\chi$-matroids by a maximality property. For trivial characters, by exploring the combinatorics of incidence stratifications, we provide a set of generators for the Chow vector spaces of the corresponding immanant varieties.


Introduction
The set Seg(k, n) of rank-one tensors in P k i=1 C n is a projective variety; it is the image under the Segre embedding of a Cartesian product of projective spaces.Given a non-zero endomorphism f of the vector space k i=1 C n , the Zariski closure of the set f (Seg(k, n)) is an irreducible algebraic set in P k i=1 C n , i.e. a projective variety (Definition 3.1 and Proposition 3.2).Under the standard action of the symmetric group S k , the tensor product k i=1 C n has the structure of a module over the group algebra C[S k ].The induced representations of simple representations of a subgroup G ⊆ S k provide idempotents of C[S k ] which are endomorphism of the tensor product above.The image of such endomorphisms are the so-called symmetry classes of tensors (see e.g.[24] for a general theory, [12] for a survey, and [23] for generalized matrix functions).
This construction allows us to associate to each simple character χ of G ⊆ S k a projective variety Gr χ (k, n) which we call immanant variety (Definition 3.17), since for G = S k the parametric equations defining it are written in terms of immanants (Theorem 3.16), which are generalizations of the determinant of a square matrix (see e.g.[29] and reference therein).In our context we use in a wider sense the word immanant for a matrix (see Definition 3.12), including the standard notions of immanants and generalized matrix functions for simple characters of finite groups (see [23]).
Among immanant varieties there is a famous one, namely the complex Grassmannian; it arises by considering the alternating character of a symmetric group and the immanant involved in the parametric equations is the determinant.In this article we prove results recovering part of the geometrical and combinatorial richness of Grassmannians for a wide class of immanant varieties.The main features explored are the following ones.
χ-matroids: it is well-known that the points of a Grassmannian Gr C (k, n) correspond to rank k matroids on the ground set [n], representable over C. A characterization of matroids, due to Gale, is by their maximality property (see [5,Theorem 1.3.1]).A maximality property can be defined in the more general case of one-dimensional characters χ of any finite group, leading to the definition of χ-matroid, see Definition 4.4.Although the points of Gr χ (k, n) are not χ-matroids in general (Example 4.7), the points of the varieties associated to trivial characters are χ-matroids (Corollary 5.4).
Incidence stratifications: the notion of incidence stratification has been introduced by the authors in [3].The stratification of a Grassmannian variety by its Schubert varieties is an example of incidence stratification (see [3,Proposition 4.16]).This construction provides a correspondence between Schubert varieties and principal order ideals of the Bruhat order on Grassmannian permutations, thanks to the maximality property of matroids.In the same vein, for the trivial character 1 G : G → {1}, the maximality property of Gr 1 G (k, n) guarantees the existence of an incidence stratification whose strata are projective varieties (Proposition 6.1 and Corollary 6.3) and whose inclusion poset is graded by dimension (Theorem 6.4) and rank-symmetric (Corollary 5.9).These stratifications are closely related to Seshadri stratifications, as recently introduced by Chirivì, Fang and Littelmann in [9].In fact, all the axioms defining Seshadri stratifications are satisfied by our stratifications, except possibly smoothness in codimension one (Example 6.5).
Chow vector spaces: it is well-known that the Chow group of the Grassmannian is free, with a basis given by the classes of Schubert varieties (see e.g.[14,Section 14.7]).By using the combinatorial results proved along the paper, we find a set of generators for the vector space obtained tensoring by Q the Chow group of Gr 1 G (k, n) (Theorem 6.4), namely the set of rational equivalence classes of the strata of the incidence stratification explained above.Moreover, applying Proposition 5.8, we give an upper bound for the Hilbert-Poincaré polynomial of the Chow vector space.
The last part of the paper is devoted to some conjectures and open problems.One of them concerns shellability of intervals in posets arising from the geometry of Gr 1 G (k, n).In parabolic quotients of Coxeter groups, the order complexes of Bruhat intervals are shellable (see [2,Theorem 2.7.5]); in particular, this holds for the Bruhat order of Grassmannian permutations, which is the inclusion poset of Schubert varieties in a Grassmannian.Since the latter is a distributive lattice, the shellability of its intervals can be deduced from a general result of Björner [1].The same can be easily proved for the intervals of the inclusion poset of the incidence strata in Gr 1 S k (k, n) (see the end of Section 5).Despite the fact that, for arbitrary groups G ⊆ S k , this poset is not a lattice in general (see Example 5.7), several experiments led us to conjecture that it is shellable (Conjecture 7.2).

Notation and preliminaries
In this section we fix notation and recall some definitions useful for the rest of the paper.We refer to [27] for posets and their incidence algebras, to [21,Chapter XVIII] for the representation theory of finite groups, to [5] and [3,Section 2.3] for matroids, to [20] and [24] for tensors, and to [14] for intersection theory.
Let Z be the ring of integer numbers, Q the field of rational numbers, R the field of real numbers, C the field of complex numbers and N the set of positive integers.For n ∈ N, we use the notation [n] := {1, 2, . . ., n}.For a finite set X, we denote by |X| its cardinality, by P(X) its power set, by X n or X ×n its n-th power under Cartesian product.If x ∈ X n , we denote by x i the projection of x on the i-th factor.If f : X → Y is a function, we let Im(f ) := {f (x) : x ∈ X}.We denote by f also the induced function f : P(X) → P(Y ).
If (X, ) is a poset, then X n is the poset given by letting x y if and only x i y i , for all i ∈ [n] and x, y ∈ X n .The set [n] is a poset under the natural order; so, for k ∈ N, the set [n] k is considered to be a poset.We denote by ⊳ a covering relation in a poset P , i.e. x ⊳ y if and only if x < y and {z ∈ P : x < z < y} = ∅.In the category of graded posets, a morphism f : X → Y is an order preserving function such that ρ 2 (f (x)) = ρ 1 (x), for all x ∈ X, where ρ 1 is the rank function of X and ρ 2 is the rank function of Y .
Let O, O Let n ∈ N and V be an n-dimensional C-vector space; define an equivalence relation ∼ on V \{0} by setting u ∼ v if and only if dim (span C {u, v}) = 1, for all u, v ∈ V \ {0}.Then, for any subset X ⊆ V , we let P(X) := π(X \ {0}), where π : V \ {0} → (V \ {0}) / ∼ is the canonical projection.In particular, P(V ) is the projective space of V .For v ∈ V \ {0}, we let [v] := π(v) ∈ P(V ).Let k ∈ N; with V ⊗k we denote the k-th tensor power of V .We let seg k,n : P(V ) ×k → P V ⊗k be the function defined by for all v 1 , . . ., v k ∈ V \ {0}.This is the so-called Segre embedding and we set Seg(k, n) We end this section by recalling the definition of incidence stratification of a projective set, as appears in [3,Section 4].Let P = ([n], P ) be a poset of cardinality n.An order ideal of P is a subset I ⊆ P such that i ∈ I and j P i imply j ∈ I.The distributive lattice of order ideals of a poset P is denoted by J (P).It is clear that there is a bijection between J (P) and the sets {max(I) : I ∈ J (P)}.For x ∈ P , we define the principal order ideal generated by x by setting x ↓ := {y ∈ P : y P x} .
For h, k ∈ N we denote by Mat h,k (C) the algebra of matrices whose entries are complex numbers.If A ∈ Mat h,k (C), i ∈ [h] and j ∈ [k], A i,j is the entry in position (i, j) of the matrix A.
Definition 2.1.The incidence algebra of P over C is The incidence group I * (P ; C) of P over C is the group of invertible elements of I(P ; C).
The subalgebra I(P ; C) ⊆ End(C n ) has invariant-subspace lattice isomorphic to J (P ), where I(P ; C) acts on the elements of C n by left multiplication.Clearly this action carries an action of I * (P ; C) on P(C n ), whose orbits are described as follows (see [3,Theorem 4.2]).
Let {e 1 , . . ., e n } be the canonical basis of C n and V I := span C {e i : i ∈ I}, for any subset I ⊆ [n].An orbit of the action of I * (P ; C) on P(C n ) is of the form for any I ∈ J (P) \{∅}, and the set of orbits {C I : I ∈ J (P) \{∅}} is a partition of P(C n ).The Zariski closure of C I is a projective space, given by for all I ∈ J (P ).
The notion of incidence stratification of a subset of a projective space, introduced in [3], includes some known affine ones, such as the stratifications given by Schubert varieties in Grassmannians and flag varieties (see [3,Propositions 4.16 and 5.5]); the following is the formal definition.Definition 2.2.Let X ⊆ P(C n ) and P be a poset of cardinality n.The set For a projective variety X, A * X denotes the group of k-cycles modulo rational equivalence on X (see [14,Chapter 1]).We let A * (X; Q) := A * X ⊗ Z Q to be the Chow vector space of X over Q.
3 The projective variety Gr f (k, n) Immanant varieties, which we are going to define in the following subsection, are particular cases of a general construction that we describe here.
Let k, n ∈ N, V be an n-dimensional complex vector space and {e 1 , . . ., e n } a basis.Let {e x : x ∈ [n] k } be the corresponding basis of V ⊗k , where e x := e x 1 ⊗ . . .⊗ e x k and x i is the projection on the i-th component of x, for all x ∈ [n] k .For f ∈ End V ⊗k , in order to introduce the main objects of our study, define a function where the overline stands for the Zariski closure.
The notation Gr f (k, n) is motivated by the fact that, for suitable choices of the function f , this construction leads to combinatorial and geometrical notions naturally appearing in the study of Grassmannian varieties.We observe that Gr f (k, n) ⊆ P (Im(f )) and that the set Im( f ) is described by a system of n k parametric polynomial equations with kn parameters.When Gr f (k, n) = ∅, i.e. f = 0, by a standard result (see e.g.[10, Proposition 4.5.5]), it is irreducible, hence a projective variety.
In several cases, the set Im( f ) is already closed in the Zariski topology, as the next result shows.
Proof.By [26, Theorem 5.2.2], given an algebraic set X ⊆ P(W ) and W a complex vector space, if F : Proof.The isomorphism is immediate since the parametric equations of Remark 3.5.For g ∈ End V ⊗k invertible, Gr f •g (k, n) and Gr f (k, n) could be not isomorphic as projective varieties.For example, let k = 2, n = 4, lex be the lexicographic order on [4] 2 , f (e x ) = e (x 1 ,x 2 ) − e (x 2 ,x 1 ) , and

Now we provide an example of a variety Gr
for all x ∈ [2] 3 .Then Im( f ) is described by the following parametric equations: With the help of Sagemath, the Gröbner basis method for implicitization by using the monomial order degrevlex (see for instance [10]) provides the following Cartesian equations for Gr f (3, 2): This is a projective variety of dimension 3, obtained as intersection of two hypersurfaces of P (Im(f )) ≃ P(C 6 ).Notice that the last two equations are the equations of We end this section by showing that some varieties Gr f (k, n) are actually Cartesian products.For h ∈ N and a ∈ N h , let a(i) := i j=1 a j , for all i ∈ [h], and k := a(h).Given The following corollary provides a realization of a product of projective spaces as a variety Gr f (k, n).

Immanant varieties
In this section we introduce the class of varieties Gr f (k, n) in which we are mostly interested, namely the ones corresponding to an endomorphism of V ⊗k arising from an action of the symmetric group S k .For k > 0, a permutation w ∈ S k induces a graded automorphism of the poset [n] k , where the action is defined by for all x ∈ [n] k .Hence w ∈ S k acts on V ⊗k by setting w(e x ) = e w(x) , for all x ∈ [n] k .This action induces an algebra morphism γ : we set P (n) := γ(P ) ∈ End V ⊗k ; hence, if P = g∈G a g g, with a g ∈ C for all g ∈ G, we have For a group G, let 1 G : G → {1} be its trivial character.If χ 1 : G → C and χ 2 : G → C are characters, their scalar product is defined by χ 1 , χ 2 := g∈G χ 1 (g)χ 2 (g −1 ).If χ is a simple character of a subgroup G ⊆ S k , then the element is an idempotent corresponding to the induced representation Ind S k G (χ) of S k .For such idempotents, we write Gr χ (k, n) instead of Gr P (n) χ (k, n).The following are well-known examples of varieties recovered in this way.
Segre embeddings: if G = {e}, the trivial subgroup of S k , there is only the trivial character χ := 1 G .In this case P Grassmannians: if G = S k and χ is its alternating character, then where sgn(σ) denotes the sign of the permutation σ.Hence, as vector spaces, Chow varieties G(1, k, n): another important class of varieties are the so-called Chow varieties G(1, k, n), the projectivization of the set of homogeneous polynomial of degree k in n variables factorizing in polynomials of degree 1.These varieties are recovered as follows.If G = S k and χ := 1 S k , then Hence, as vector spaces, Im P is the Chow variety G(1, k, n).For more details see e.g.[16,Chapter 4] and [20,Section 8.6].
The following construction realizes a Cartesian product of projective spaces differently with respect to Corollary 3.8.
Cartesian product of projective spaces: let h ∈ N, a ∈ N h and k := a(h), where a(j) := j i=1 a i , for all j ∈ [h].The symmetric group S k is generated by the simple transpositions {s 1 , . . ., s k−1 }; for i, j ∈ [k − 1], i j, define the parabolic subgroup We define the parabolic subgroup G a ⊆ S k by Hence is the idempotent corresponding to the Young module M a (see e.g.[7, Section 3.6.2]).Since, for k 1, the Chow variety G(1, k, 2) is isomorphic to P(C k+1 ) (because any homogeneous polynomial of positive degree in two variables factorizes as product of degree one polynomials), we obtain For a simple character . We denote by Res G H (χ) the restriction to a group H ⊆ G of a character χ of G. Given an action of a group G on a set X, let G x be the isotropy group of the element x ∈ X, i.e.G x := {g ∈ G : g(x) = x}.
A set of generators and the dimension of the vector space V ⊗k χ are known, see [24, Eq. 6.13 and 6.23].
Theorem 3.9.Let G ⊆ S k be a subgroup and χ a simple character of G. Then P (n) χ (e x ) = 0 if and only if Res G Gx (χ), 1 Gx = 0, for every x ∈ [n] k , and where c(g) is the number of cycles of the permutation g ∈ S k .

Parametric equations for Im
where (a 1 , a 2 ), (b 1 , b 2 ), (c 1 , c 2 ) ∈ C 2 \{(0, 0)}.Using Gröbner basis method for implicitization, we find the following Cartesian equations for Gr χ (3, 2): These equations are the ones for Gr χ (3, 2); in fact it can be checked by hand that P (2)  χ (e (1,1,2) ) − P (2)  χ (e (2,1,1) In the following, we are going to introduce a generalization of the immanant of a square matrix to matrices with arbitrary size, depending on a subgroup G ⊆ S k and a simple character of G. Our definition extends to submatrices the notion of generalized matrix function (see [24,Chapter 7]).
Let k, m, n ∈ N; recall that V is an n-dimensional complex vector space.If W is an m-dimensional complex vector space and f ∈ Hom(V, W ), the element f ⊗k ∈ Hom(V ⊗k , W ⊗k ) is defined by setting Remark 3.11.When W = V , we have that (f • g) ⊗k = f ⊗k • g ⊗k , for all f, g ∈ End(V ), and (Id V ) ⊗k = Id V ⊗k , i.e. the function f → f ⊗k defines a monoid morphism End(V ) → End(V ⊗k ).

Given a matrix
with respect to bases of V and W , the matrix associated to f ⊗k is M ⊗k .In the following definition we extend the notion of immanant of a square matrix.Let M ∈ Mat n,n (C), k = n and x = y = (1, . . ., n) =: n.In this setting, for G = S n , we recover some well-known numbers associated to M .
• If χ is the alternating character of S n , then χ n, n (M ) is the determinant of the matrix M .
• If χ := 1 Sn , then χ n, n (M ) is the permanent of the matrix M .
• If χ is any simple character of S n , then χ n, n (M ) is the so-called immanant of the matrix M , see for instance [29] and references therein.
The χ x,y -immanant of a matrix M is related to the (x, y)-entry of the matrix M ⊗k P (n) χ , as the next proposition asserts.
Proof.Let y ∈ [n] k and {w i : i ∈ [m]} be a basis of the m-dimensional vector space W .We have that Remark 3.15.For P ∈ C[S k ], we can define, by restriction, an element f ⊗k P ∈ Hom Im(P (n) ), Im(P (m) ) ; if P = P χ for some simple character χ of a group G ⊆ S k , then f ⊗k P ∈ Hom C[G] Im(P (n) ), Im(P (m) ) , since gP χ = P χ g, for all g ∈ G, and then Im(P (n) ), Im(P (m) ) are C[G]-modules.See [22] for a more extended treatment on such induced morphisms.Now we provide parametric equations for the set Im P where none of the columns is the zero vector.
Theorem 3.16.Let k, n ∈ N and χ be a simple character of a group G ⊆ S k .Then the set Im( P (n) χ ) ⊆ P V ⊗k is described by the parametric equations where A is the generic matrix defined above.
Proof.Let v i = a 1i e 1 + . . .+ a ni e n , for all i ∈ [k], and let W be a vector space with basis {ẽ 1 , . . ., ẽk }.Then A ∈ Hom(W, V ) and, by Proposition 3.14, The statement of Theorem 3.16 leads us to the following definition.
Definition 3.17.If χ is a simple character of a group G ⊆ S k , we call Gr χ (k, n) an immanant variety.
We observe that, if two subgroups G ⊆ S k and H ⊆ S k are conjugated in S k , i.e.H = σGσ −1 for some σ ∈ S k , and χ is a character of G, then, as projective varieties, Gr χ (k, n) ≃ Gr χ σ (k, n), where χ σ (h) := χ(σ −1 hσ), for all h ∈ H.In fact by Proposition 3.4.In the following example we see that . Hence it is a smooth four-dimensional projective variety.On the other hand, we checked by using Macaulay2 [17] that Gr 1 H (4, 2) is a singular four-dimensional projective varieties.Then, as projective varieties, they are not isomorphic.

One-dimensional characters and χ-matroids
In this section we restrict our attention to one-dimensional characters of a subgroup G ⊆ S k .This includes, for example, the trivial character of G and all the simple characters of an abelian group. For k and lex the lexicographic order on O x .Then we define For completeness, we give a proof of the following theorem, ensuring that, for one-dimensional characters, the spanning set of Theorem 3.9 provides a basis in a canonical way.It is known in another formulation, see [24,Corollary 6.32].
Theorem 4.1.Let G ⊆ S k be a subgroup and χ a one-dimensional character of G, i.e. a group morphism χ : G → C \{0}.Then is a basis of V ⊗k χ .
For example, if G = {e} is the trivial group, then n of Grassmannian permutations with the Bruhat order (see [3,Proposition 4.9]).
where µ is the Möbius function.For more information on Lyndon words, see e.g.[28,Exercise 7.89].The following is the Hasse diagram of B χ (6, 2). ( Now we extend the notion of rank k matroids to all one-dimensional characters χ of a group G ⊆ S k .Let σ ∈ S n ; then σ acts on [n] k by letting σ * (x) := (σ(x 1 ), . . ., σ(x k )), for all x ∈ [n] k .Hence we have an action of S n on V ⊗k , commuting with the previously defined action of S k on V ⊗k .Definition 4.4.Let X ⊆ B χ (k, n).We say that X is a χ-matroid if, for every σ ∈ S n , the induced subposet σ * (x) : x ∈ X ⊆ B χ (k, n) has a unique maximum, where for x ∈ [n] k , x is defined in (2).Remark 4.5.A characterization of matroids, due to Gale, is by their maximality property (see [5,Theorem 1.3.1]).Therefore, if χ is the alternating character of G = S k , we recover the set of bases of a rank k matroid on the ground set [n].
Several important varieties have the maximality property.For example, the Grassmannian Gr C (k, n) has this property.In fact, the elements of Gr C (k, n) correspond to representable matroids of rank k and matroids can be defined by their maximality property ( [5,Theorem 1.3.1]).This is also the case for Segre varieties.Proof.For k ∈ N, let e(k) be the identity in S k .The statement is equivalent to say that supp and this concludes the proof.
More in general, when χ is a trivial character, the set Im P (n) χ has the maximality property, as we prove in the next section (see Corollary 5.4).We end this section with one more definition.Definition 4.9.We say that a subset has the maximality property if and only all representable subsets X ⊆ B χ (k, n) are χ-matroids.Differently to the Grassmannian case, there exist representable subsets in B χ (k, n) which are not χ-matroids, see Example 4.7.
and the variety Gr χ (k, n) has the maximality property, then the action on P V ⊗k χ of the group of invertible diagonal matrices of size m (i.e. the incidence group of the trivial poset of cardinality m), provides an incidence stratification of Gr χ (k, n) whose strata are in bijection with representable χ-matroids.If χ is the alternating character of S k , we recover the matroidal strata introduced in [15].These strata provide a geometric interpretation of the Tutte polynomials via the K-theory of Grassmannians, see [13].
, because the action of S k on [n] k preserves its rank.This implies P (n) (v) = 0; hence Seg(k, n)∩ P ker P (n) = ∅ and the result follows by Proposition 3.3.
Notice that, in the trivial character case, we have the following commutative diagram of functions, where ∼ G is the equivalence relation on P(V ) ×k defined by setting ), for some g ∈ G, and π G is the canonical projection on the quotient.
The function seg k,n G is the unique one such that P (n) It is an injective function by [24,Theorem 6.60 Our next aim is to prove that the variety Gr 1 G (k, n) has the maximality property.We define an idempotent function p 1 Corollary 5.4.The set Gr 1 G (k, n) has the maximality property.In particular, Proof.The poset B 1 G (k, n) has maximum and minimum, (n, n, . . ., n) and (1, 1, . . ., 1), respectively.Let x ⊳ y in B 1 G (k, n).Then x g(y) for some g ∈ G. Let z ∈ [n] k be such that x < z < y.By Proposition 5.3 we have that Remark 5.6.Notice that Proposition 5.5 follows directly by [18,Lemma 7] since B 1 G (k, n) is a homogeneous quotient of the poset B 1 {e} (k, n).Moreover, by Propositions 5.3 and 5.5 it follows that p 1 G is a morphism of graded posets.
In the following example we depict the Hasse diagram of one of these posets.
Proposition 5.8.The rank-generating function of the poset [n] q i c i (g) , where [n] q := n−1 i=0 q i is the q-analog of n and c i (g) is the number of cycles of length i ∈ [k] of the permutation g ∈ S k .
Proof.The result follows by using the weighted Pólya enumeration theorem with [n] as set of colours and weights w(i) = i − 1, for all i ∈ [n].Then the generating function of the colours is [n] q and ρ(x) = k i=1 w(x i ), for all x ∈ B 1 G (k, n).Therefore the generating function of the number of colored arrangements by weight is the rank-generating function of B 1 G (k, n).Corollary 5.9.Let G ⊆ S k be a group; then the poset B 1 G (k, n) is ranksymmetric.
6 The geometry of Gr 1 G (k, n) In this section we use ideas from [3] and the combinatorial tools of Section 5 to realize an incidence stratification of the variety Gr 1 G (k, n).This stratification provides a set of generators for the Chow Q-vector space of Gr 1 G (k, n), as we prove in Theorem 6.4.We consider the incidence group over C of the poset B 1 G (k, n) acting on the projective space P V ⊗k 1 G .Then, by [3,Theorem 5.1] the orbits of this action are in bijection with the order ideals of B 1 G (k, n).Given an order ideal I ∈ J (B 1 G (k, n)) let C I be the corresponding orbit and define Proposition 6.1.We have that C 1 G I = ∅ if and only if I is a principal order ideal.Hence as posets Proof.Let I be principal with m := max(I).Then P x ↓ .By Proposition 6.1, it follows that Let x ∈ [n] k and A ∈ Mat n,k (C) be the matrix of parameters (see Theorem 3.16).We let A x ∈ Mat n,k (C) be the matrix whose entries are A x i,j = A i,j if i x j , and A x i,j = 0 otherwise.We now provide a system of parametric equations for the algebraic set C x ↓ ∩ Gr 1 G (k, n).Along the paper, we define χ-matroids for a one-dimensional character χ of a group G ⊆ S k (see Definition 4.4).We believe that this notion has an independent interest.Problem 7.5.Find crypthomorphic definitions of χ-matroids and explore their combinatorial properties.

Definition 3 . 12 .
Let x ∈ [m] k , y ∈ [n] k and χ a simple character of a group G ⊆ S k .The χ x,y -immanant of a matrix M ∈ Mat m,n (C) is defined by χ x,y (M ) := g∈G χ(g) M ⊗k g(x),y .

5
The combinatorics of Gr 1 G (k, n)In this section we consider the trivial character1 G of a subgroup G ⊆ S k .In this case B 1 G (k, n) = {x : x ∈ [n] k } because the condition G x ⊆ ker(1 G ) = G (see Theorem4.1) is trivially satisfied.Moreover we have that P ker P (n) 1 G ∩ Seg(k, n) = ∅; in fact we can state the following more general result.Proposition 5.1.Let P ∈ C[S k ] be such that P

Proposition 5 . 5 .
), for all x ∈ [n] k .The restriction of ρ provides the rank function of the poset B 1 G (k, n), as the next result shows.The poset B 1 G (k, n) is graded with rank function ρ.

(n) 1 G
(e m ) ∈ C I ∩ Gr 1 G (k, n).On the other hand, let [v] ∈ C I ∩ Gr 1 G (k, n) for some v ∈ V ⊗k 1 G .By Corollary 5.4, we have that max(I) = max(supp 1 G [v]) = {m} for some m ∈ B 1 G (k, n).The rest of the statement is an immediate consequence.For x ∈ B 1 G (k, n), we set C x := C 1 G
1 and O 2 be objects in a category.The notation Hom(O 1 , O 2 ) stands for the set of morphisms between O 1 and O 2 .We let End(O) := Hom(O, O) and O 1 ≃ O 2 denotes the existence of an isomorphism.
Example 3.13.Let M ∈ Mat 2,3 (C) be the generic matrix M = a 11 a 12 a 13 a 21 a 22 a 23 .11 a 12 a 11 a 13 a 11 a 12 a 2 12 a 12 a 13 a 11 a 13 a 12 a 13 a 2 13 a 11 a 21 a 11 a 22 a 11 a 23 a 12 a 21 a 12 a 22 a 12 a 23 a 13 a 21 a 13 a 22 a 13 a 23 a 21 a 11 a 21 a 12 a 21 a 13 a 22 a 11 a 22 a 12 a 22 a 13 a 23 a 11 a 23 a 12 a 23 a 13 a 2 21 a 21 a 22 a 21 a 23 a 22 a 21 a 2 22 a 22 a 23 a 23 a 21 a 23 a 22