# Immanant varieties

Davide Bolognini and Paolo Sentinelli Dipartimento di Ingegneria Industriale e
Scienze Matematiche, Università Politecnica delle Marche, Ancona, Italy.
<EMAIL_ADDRESS>Dipartimento di Matematica, Politecnico di
Milano, Milan, Italy.
<EMAIL_ADDRESS>
###### Abstract

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.

Keywords: _immanant, matroid, symmetric group, one-dimensional characters,
Chow group_

Math. sub class: _05B35, 05E45, 05E18, 06A07, 15A69, 20B30_

##  1 Introduction

The set $\operatorname{\mathrm{Seg}}(k,n)$ of rank-one tensors in
$\operatorname{\mathbb{P}}\left(\bigotimes_{i=1}^{k}\operatorname{\mathbb{C}}^{n}\right)$
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 $\bigotimes_{i=1}^{k}\operatorname{\mathbb{C}}^{n}$, the
Zariski closure of the set $f(\operatorname{\mathrm{Seg}}(k,n))$ is an
irreducible algebraic set in
$\operatorname{\mathbb{P}}\left(\bigotimes_{i=1}^{k}\operatorname{\mathbb{C}}^{n}\right)$,
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
$\bigotimes_{i=1}^{k}\operatorname{\mathbb{C}}^{n}$ has the structure of a
module over the group algebra $\operatorname{\mathbb{C}}[S_{k}]$. The induced
representations of simple representations of a subgroup $G\subseteq S_{k}$
provide idempotents of $\operatorname{\mathbb{C}}[S_{k}]$ 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 $\chi$ of
$G\subseteq S_{k}$ a projective variety
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)$ 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.

$\chi$-matroids: it is well-known that the points of a Grassmannian
$\operatorname{\mathrm{Gr}}_{\operatorname{\mathbb{C}}}(k,n)$ correspond to
rank $k$ matroids on the ground set $[n]$, representable over
$\operatorname{\mathbb{C}}$. 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 $\chi$
of any finite group, leading to the definition of $\chi$-matroid, see
Definition 4.4. Although the points of
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)$ are not $\chi$-matroids in general
(Example 4.7), the points of the varieties associated to trivial characters
are $\chi$-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 $\mathbf{1}_{G}:G\rightarrow\\{1\\}$,
the maximality property of $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$
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
$\mathbb{Q}$ the Chow group of
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ (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 $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$. 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 $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{S_{k}}}(k,n)$
(see the end of Section 5). Despite the fact that, for arbitrary groups
$G\subseteq S_{k}$, 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).

##  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 $\operatorname{\mathbb{Z}}$ be the ring of integer numbers,
$\operatorname{\mathbb{Q}}$ the field of rational numbers,
$\operatorname{\mathbb{R}}$ the field of real numbers,
$\operatorname{\mathbb{C}}$ the field of complex numbers and $\mathbb{N}$ the
set of positive integers. For $n\in\mathbb{N}$, we use the notation
$[n]:=\left\\{1,2,\ldots,n\right\\}$. For a finite set $X$, we denote by $|X|$
its cardinality, by $\mathcal{P}(X)$ its power set, by $X^{n}$ or $X^{\times
n}$ its $n$-th power under Cartesian product. If $x\in X^{n}$, we denote by
$x_{i}$ the projection of $x$ on the $i$-th factor. If $f:X\rightarrow Y$ is a
function, we let $\operatorname{\mathrm{Im}}(f):=\\{f(x):x\in X\\}$. We denote
by $f$ also the induced function $f:\mathcal{P}(X)\rightarrow\mathcal{P}(Y)$.

If $(X,\preccurlyeq)$ is a poset, then $X^{n}$ is the poset given by letting
$x\leqslant y$ if and only $x_{i}\preccurlyeq y_{i}$, for all $i\in[n]$ and
$x,y\in X^{n}$. The set $[n]$ is a poset under the natural order; so, for
$k\in\operatorname{\mathbb{N}}$, the set $[n]^{k}$ is considered to be a
poset. We denote by $\operatorname{\vartriangleleft}$ a covering relation in a
poset $P$, i.e. $x\operatorname{\vartriangleleft}y$ if and only if $x<y$ and
$\\{z\in P:x<z<y\\}=\varnothing$. In the category of graded posets, a morphism
$f:X\rightarrow Y$ is an order preserving function such that
$\rho_{2}(f(x))=\rho_{1}(x)$, for all $x\in X$, where $\rho_{1}$ is the rank
function of $X$ and $\rho_{2}$ is the rank function of $Y$.

Let $O,O_{1}$ and $O_{2}$ be objects in a category. The notation
$\operatorname{\mathrm{Hom}}(O_{1},O_{2})$ stands for the set of morphisms
between $O_{1}$ and $O_{2}$. We let
$\operatorname{\mathrm{End}}(O):=\operatorname{\mathrm{Hom}}(O,O)$ and
$O_{1}\simeq O_{2}$ denotes the existence of an isomorphism.

Let $n\in\operatorname{\mathbb{N}}$ and $V$ be an $n$-dimensional
$\operatorname{\mathbb{C}}$-vector space; define an equivalence relation
$\sim$ on $V\setminus\\{0\\}$ by setting $u\sim v$ if and only if
$\dim\left(\operatorname{span}_{\operatorname{\mathbb{C}}}\\{u,v\\}\right)=1$,
for all $u,v\in V\setminus\\{0\\}$. Then, for any subset $X\subseteq V$, we
let $\operatorname{\mathbb{P}}(X):=\pi(X\setminus\\{0\\})$, where
$\pi:V\setminus\\{0\\}\rightarrow\left(V\setminus\\{0\\}\right)/\sim$ is the
canonical projection. In particular, $\operatorname{\mathbb{P}}(V)$ is the
projective space of $V$. For $v\in V\setminus\\{0\\}$, we let
$[v]:=\pi(v)\in\operatorname{\mathbb{P}}(V)$. Let
$k\in\operatorname{\mathbb{N}}$; with $V^{\otimes k}$ we denote the $k$-th
tensor power of $V$. We let
$\mathrm{seg}^{k,n}:\operatorname{\mathbb{P}}(V)^{\times
k}\rightarrow\operatorname{\mathbb{P}}\left(V^{\otimes k}\right)$ be the
function defined by

$\mathrm{seg}^{k,n}([v_{1}],\ldots,[v_{k}])=[v_{1}\otimes\ldots\otimes
v_{k}],$

for all $v_{1},\ldots,v_{k}\in V\setminus\\{0\\}$. This is the so-called
_Segre embedding_ and we set
$\operatorname{\mathrm{Seg}}(k,n):=\operatorname{\mathrm{Im}}(\mathrm{seg}^{k,n})$.
It is well-known that $\operatorname{\mathrm{Seg}}(k,n)$ is a
$k(n-1)$-dimensional projective subvariety of
$\operatorname{\mathbb{P}}\left(V^{\otimes k}\right)$.

We end this section by recalling the definition of incidence stratification of
a projective set, as appears in [3, Section 4]. Let
$P=\left([n],\leqslant_{P}\right)$ be a poset of cardinality $n$. An order
ideal of $P$ is a subset $I\subseteq P$ such that $i\in I$ and
$j\leqslant_{P}i$ imply $j\in I$. The distributive lattice of order ideals of
a poset $P$ is denoted by $\operatorname{\mathcal{J}(P)}$. It is clear that
there is a bijection between $\operatorname{\mathcal{J}(P)}$ and the sets
$\left\\{\max(I):I\in\operatorname{\mathcal{J}(P)}\right\\}$. For $x\in P$, we
define the principal order ideal generated by $x$ by setting

$x^{\downarrow}:=\left\\{y\in P:y\leqslant_{P}x\right\\}.$

For $h,k\in\operatorname{\mathbb{N}}$ we denote by
$\mathrm{Mat}_{h,k}(\operatorname{\mathbb{C}})$ the algebra of matrices whose
entries are complex numbers. If
$A\in\mathrm{Mat}_{h,k}(\operatorname{\mathbb{C}})$, $i\in[h]$ and $j\in[k]$,
$A_{i,j}$ is the entry in position $(i,j)$ of the matrix $A$.

######  Definition 2.1.

The incidence algebra of $P$ over $\operatorname{\mathbb{C}}$ is

$I(P;\operatorname{\mathbb{C}}):=\left\\{A\in\mathrm{Mat}_{n,n}(\operatorname{\mathbb{C}}):i\nleqslant_{P}j\Rightarrow
A_{i,j}=0\right\\}.$

The _incidence group_ $I^{*}(P;\operatorname{\mathbb{C}})$ of $P$ over
$\operatorname{\mathbb{C}}$ is the group of invertible elements of
$I(P;\operatorname{\mathbb{C}})$.

The subalgebra
$I(P;\operatorname{\mathbb{C}})\subseteq\operatorname{\mathrm{End}}(\operatorname{\mathbb{C}}^{n})$
has invariant-subspace lattice isomorphic to $\mathcal{J}(P)$, where
$I(P;\operatorname{\mathbb{C}})$ acts on the elements of
$\operatorname{\mathbb{C}}^{n}$ by left multiplication. Clearly this action
carries an action of $I^{*}(P;\operatorname{\mathbb{C}})$ on
$\mathbb{P}(\operatorname{\mathbb{C}}^{n})$, whose orbits are described as
follows (see [3, Theorem 4.2]).

Let $\\{e_{1},\ldots,e_{n}\\}$ be the canonical basis of
$\operatorname{\mathbb{C}}^{n}$ and
$V_{I}:=\operatorname{span}_{\operatorname{\mathbb{C}}}\left\\{e_{i}:i\in
I\right\\}$, for any subset $I\subseteq[n]$. An orbit of the action of
$I^{*}(P;\operatorname{\mathbb{C}})$ on
$\mathbb{P}(\operatorname{\mathbb{C}}^{n})$ is of the form

$C_{I}:=\mathbb{P}(V_{I})\setminus\bigcup\limits_{i\in\mathrm{max}(I)}\mathbb{P}\left(V_{I\setminus\\{i\\}}\right),$

for any $I\in\operatorname{\mathcal{J}(P)}\setminus\\{\varnothing\\}$, and the
set of orbits
$\\{C_{I}:I\in\operatorname{\mathcal{J}(P)}\setminus\\{\varnothing\\}\\}$ is a
partition of $\mathbb{P}(\operatorname{\mathbb{C}}^{n})$. The Zariski closure
of $C_{I}$ is a projective space, given by

$\overline{C}_{I}=\biguplus\limits_{H\in\mathcal{J}(I)\setminus\\{\varnothing\\}}C_{H}=\mathbb{P}\left(V_{I}\right),$

for all $I\in\mathcal{J}(P)$.

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 $X\subseteq\mathbb{P}(\operatorname{\mathbb{C}}^{n})$ and $P$ be a poset
of cardinality $n$. The set

$\left\\{\overline{C}_{I}\cap
X:I\in\mathcal{J}(P)\right\\}\setminus\left\\{\varnothing\right\\}$

is an _incidence stratification_ of $X$.

For a projective variety $X$, $A_{*}X$ denotes the group of $k$-_cycles modulo
rational equivalence_ on $X$ (see [14, Chapter 1]). We let
$A_{*}(X;\operatorname{\mathbb{Q}}):=A_{*}X\otimes_{\operatorname{\mathbb{Z}}}\operatorname{\mathbb{Q}}$
to be the _Chow vector space_ of $X$ over $\operatorname{\mathbb{Q}}$.

##  3 The projective variety $\mathrm{Gr}_{f}(k,n)$

Immanant varieties, which we are going to define in the following subsection,
are particular cases of a general construction that we describe here.

Let $k,n\in\operatorname{\mathbb{N}}$, $V$ be an $n$-dimensional complex
vector space and $\\{e_{1},\ldots,e_{n}\\}$ a basis. Let
$\\{e_{x}:x\in[n]^{k}\\}$ be the corresponding basis of $V^{\otimes k}$, where
$e_{x}:=e_{x_{1}}\otimes\ldots\otimes e_{x_{k}}$ and $x_{i}$ is the projection
on the $i$-th component of $x$, for all $x\in[n]^{k}$. For
$f\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$, in order to
introduce the main objects of our study, define a function

$\widehat{f}:\operatorname{\mathrm{Seg}}(k,n)\setminus\operatorname{\mathbb{P}}(\ker(f))\rightarrow\operatorname{\mathbb{P}}\left(V^{\otimes
k}\right)$

by setting $\widehat{f}([v_{1}\otimes\ldots\otimes
v_{k}])=\left[f(v_{1}\otimes\ldots\otimes v_{k})\right]$, for all
$v_{1},\ldots,v_{k}\in V\setminus\\{0\\}$ such that
$f(v_{1}\otimes\ldots\otimes v_{k})\neq 0$.

######  Definition 3.1.

Let $f\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$. The algebraic
set
$\operatorname{\mathrm{Gr}}_{f}(k,n)\subseteq\operatorname{\mathbb{P}}(V^{\otimes
k})$ is defined by setting

$\operatorname{\mathrm{Gr}}_{f}(k,n):=\overline{\operatorname{\mathrm{Im}}(\,\widehat{f}\,)},$

where the overline stands for the Zariski closure.

The notation $\operatorname{\mathrm{Gr}}_{f}(k,n)$ 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
$\operatorname{\mathrm{Gr}}_{f}(k,n)\subseteq\operatorname{\mathbb{P}}\left(\operatorname{\mathrm{Im}}(f)\right)$
and that the set $\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$ is described by
a system of $n^{k}$ parametric polynomial equations with $kn$ parameters. When
$\operatorname{\mathrm{Gr}}_{f}(k,n)\neq\varnothing$, i.e. $f\neq 0$, by a
standard result (see e.g. [10, Proposition 4.5.5]), it is irreducible, hence a
projective variety.

######  Proposition 3.2.

Let $f\in\operatorname{\mathrm{End}}\left(V^{\otimes
k}\right)\setminus\\{0\\}$; then the algebraic set
$\operatorname{\mathrm{Gr}}_{f}(k,n)$ is a projective variety.

In several cases, the set $\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$ is
already closed in the Zariski topology, as the next result shows.

######  Proposition 3.3.

Let $f\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$; then

$\operatorname{\mathrm{Seg}}(k,n)\cap\operatorname{\mathbb{P}}(\ker(f))=\varnothing\,\,\Rightarrow\,\,\operatorname{\mathrm{Gr}}_{f}(k,n)=\operatorname{\mathrm{Im}}(\,\widehat{f}\,).$

###### Proof.

By [26, Theorem 5.2.2], given an algebraic set
$X\subseteq\operatorname{\mathbb{P}}(W)$ and $W$ a complex vector space, if
$F:X\rightarrow\operatorname{\mathbb{P}}(W)$ is a polynomial function, then
$F$ is closed. Since
$\operatorname{\mathrm{Seg}}(k,n)\cap\operatorname{\mathbb{P}}(\ker(f))=\varnothing$,
it holds that $\widehat{f}$ is a polynomial function from
$\operatorname{\mathrm{Seg}}(k,n)$ to
$\operatorname{\mathbb{P}}\left(V^{\otimes k}\right)$. From the fact that
$\operatorname{\mathrm{Seg}}(k,n)$ is a projective variety it follows that
$\widehat{f}$ is closed. ∎

The simplest example of these varieties corresponds to
$f=\operatorname{\mathrm{Id}}_{V^{\otimes k}}$; in this case
$\operatorname{\mathrm{Im}}(\,\widehat{f}\,)=\operatorname{\mathrm{Gr}}_{f}(k,n)=\operatorname{\mathrm{Seg}}(k,n)$.
By using Proposition 3.3, it is not difficult to show that if
$f\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$ is invertible,
then
$\operatorname{\mathrm{Gr}}_{f}(k,n)\simeq\operatorname{\mathrm{Seg}}(k,n)$ as
projective varieties. The general situation is summarized in the following
result.

######  Proposition 3.4.

Let $f,g\in\operatorname{\mathrm{End}}\left(V^{\otimes
k}\right)\setminus\\{0\\}$ with $g$ invertible. Then
$\operatorname{\mathrm{Gr}}_{g\circ
f}(k,n)\simeq\operatorname{\mathrm{Gr}}_{f}(k,n)$, as projective varieties.

###### Proof.

The isomorphism is immediate since the parametric equations of
$\operatorname{\mathrm{Gr}}_{g\circ f}(k,n)$ in the basis
$\\{g(e_{x}):x\in[n]^{k}\\}$ are the ones of
$\operatorname{\mathrm{Gr}}_{f}(k,n)$ in the basis $\\{e_{x}:x\in[n]^{k}\\}$.
∎

######  Remark 3.5.

For $g\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$ invertible,
$\operatorname{\mathrm{Gr}}_{f\circ g}(k,n)$ and
$\operatorname{\mathrm{Gr}}_{f}(k,n)$ could be not isomorphic as projective
varieties. For example, let $k=2$, $n=4$,
$\leqslant_{\operatorname{\mathrm{lex}}}$ be the lexicographic order on
$[4]^{2}$, $f(e_{x})=e_{(x_{1},x_{2})}-e_{(x_{2},x_{1})}$, and

$g(e_{x})=\left\\{\begin{array}[]{ll}\sum\limits_{y\leqslant_{\operatorname{\mathrm{lex}}}x}e_{y},&\hbox{if
$x\leqslant_{\operatorname{\mathrm{lex}}}(3,1)$;}\\\
e_{x},&\hbox{otherwise,}\end{array}\right.$

for all $x\in[4]^{2}$. Then $\operatorname{\mathrm{Gr}}_{f}(2,4)$ is the
complex Grassmannian
$\operatorname{\mathrm{Gr}}_{\operatorname{\mathbb{C}}}(2,4)$ and
$\operatorname{\mathrm{Gr}}_{f\circ
g}(2,4)=\operatorname{\mathbb{P}}\left(\operatorname{\mathrm{Im}}(f)\right)\simeq\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{6})$.

Now we provide an example of a variety $\operatorname{\mathrm{Gr}}_{f}(k,n)$
corresponding to a not invertible endomorphism.

######  Example 3.6.

Let $k=3$, $n=2$ and $f\in\operatorname{\mathrm{End}}\left(V^{\otimes
3}\right)$ defined by setting

$f(e_{x})=\left\\{\begin{array}[]{ll}e_{(2,1,2)}+e_{(2,2,1)}+e_{(2,2,2)},&\hbox{if
$x\in\left\\{(2,1,2),(2,2,1),(2,2,2)\right\\}$;}\\\
e_{x},&\hbox{otherwise,}\end{array}\right.$

for all $x\in[2]^{3}$. Then $\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$ is
described by the following parametric equations:

$\left\\{\begin{array}[]{ll}x_{111}=a_{1}b_{1}c_{1}\\\
x_{112}=a_{1}b_{1}c_{2}\\\ x_{121}=a_{1}b_{2}c_{1}\\\
x_{122}=a_{1}b_{2}c_{2}\\\ x_{211}=a_{2}b_{1}c_{1}\\\
x_{212}=a_{2}b_{1}c_{2}+a_{2}b_{2}c_{1}+a_{2}b_{2}c_{2}\\\
x_{221}=a_{2}b_{1}c_{2}+a_{2}b_{2}c_{1}+a_{2}b_{2}c_{2}\\\
x_{222}=a_{2}b_{1}c_{2}+a_{2}b_{2}c_{1}+a_{2}b_{2}c_{2}\end{array}\right.$

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 $\operatorname{\mathrm{Gr}}_{f}(3,2)$:

$\left\\{\begin{array}[]{ll}(x_{112}+x_{121}+x_{122})x_{211}-x_{111}x_{222}=0\\\
x_{112}x_{121}-x_{111}x_{122}=0\\\ x_{212}-x_{222}=0\\\
x_{221}-x_{222}=0\end{array}\right.$

This is a projective variety of dimension $3$, obtained as intersection of two
hypersurfaces of
$\operatorname{\mathbb{P}}\left(\operatorname{\mathrm{Im}}(f)\right)\simeq\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{6})$.
Notice that the last two equations are the equations of
$\operatorname{\mathbb{P}}\left(\operatorname{\mathrm{Im}}(f)\right)$ in
$\operatorname{\mathbb{P}}\left(V^{\otimes 3}\right)$. The set
$\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$ is not Zariski-closed. In fact,
if $v:=e_{(1,2,2)}+e_{(2,1,1)}-e_{(1,2,2)}$, then
$[v]\in\operatorname{\mathrm{Gr}}_{f}(3,2)\setminus\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$.

We end this section by showing that some varieties
$\operatorname{\mathrm{Gr}}_{f}(k,n)$ are actually Cartesian products. For
$h\in\operatorname{\mathbb{N}}$ and $a\in\operatorname{\mathbb{N}}^{h}$, let
$a(i):=\sum_{j=1}^{i}a_{j}$, for all $i\in[h]$, and $k:=a(h)$. Given
$f_{i}\in\operatorname{\mathrm{End}}\left(V^{\otimes a_{i}}\right)$, for all
$i\in[h]$, we define $f_{1}\otimes\ldots\otimes
f_{h}\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$ by setting

$(f_{1}\otimes\ldots\otimes
f_{h})(e_{x}):=f_{1}(e_{x_{[1,a(1)]}})\otimes\ldots\otimes
f_{h}(e_{x_{[a(h-1)+1,a(h)]}}),$

for all $x\in[n]^{k}$, where, for $i,j\in[k]$, $i\leqslant j$,
$x_{[i,j]}:=(x_{i},\ldots,x_{j})\in[n]^{j-i+1}$.

######  Proposition 3.7.

Let $h\in\operatorname{\mathbb{N}}$, $a\in\operatorname{\mathbb{N}}^{h}$ and
$k:=a(h)$. Let $f_{i}\in\operatorname{\mathrm{End}}\left(V^{\otimes
a_{i}}\right)\setminus\\{0\\}$, for all $i\in[h]$. Then, as projective
varieties,

$\operatorname{\mathrm{Gr}}_{f_{1}\otimes\ldots\otimes
f_{h}}(k,n)\simeq\operatorname{\mathrm{Gr}}_{f_{1}}(a_{1},n)\times\ldots\times\operatorname{\mathrm{Gr}}_{f_{h}}(a_{h},n).$

The following corollary provides a realization of a product of projective
spaces as a variety $\operatorname{\mathrm{Gr}}_{f}(k,n)$.

######  Corollary 3.8.

Let $f_{1},\ldots,f_{k}\in\operatorname{\mathrm{End}}(V)\setminus\\{0\\}$.
Then, as projective varieties,

$\operatorname{\mathrm{Gr}}_{f_{1}\otimes\ldots\otimes
f_{k}}(k,n)\simeq\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{b_{1}})\times\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{b_{2}})\times\ldots\times\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{b_{k}}),$

where $b_{i}:=\dim\left(\operatorname{\mathrm{Im}}(f_{i})\right)$, for all
$i\in[k]$.

###  3.1 Immanant varieties

In this section we introduce the class of varieties
$\operatorname{\mathrm{Gr}}_{f}(k,n)$ in which we are mostly interested,
namely the ones corresponding to an endomorphism of $V^{\otimes k}$ arising
from an action of the symmetric group $S_{k}$. For $k>0$, a permutation $w\in
S_{k}$ induces a graded automorphism of the poset $[n]^{k}$, where the action
is defined by

$w(x)=\left(x_{w^{-1}(1)},\ldots,x_{w^{-1}(k)}\right),$

for all $x\in[n]^{k}$. Hence $w\in S_{k}$ acts on $V^{\otimes k}$ by setting
$w(e_{x})=e_{w(x)}$, for all $x\in[n]^{k}$. This action induces an algebra
morphism
$\gamma:\operatorname{\mathbb{C}}[S_{k}]\rightarrow\operatorname{\mathrm{End}}\left(V^{\otimes
k}\right)$, where $\operatorname{\mathbb{C}}[S_{k}]$ is the group algebra over
$\operatorname{\mathbb{C}}$ of $S_{k}$.

Let $G\subseteq S_{k}$ be a group. It is clear that the group algebra
$\operatorname{\mathbb{C}}[G]$ is a subalgebra of
$\operatorname{\mathbb{C}}[S_{k}]$. For
$P\in\operatorname{\mathbb{C}}[S_{k}]$, we set
$P^{(n)}:=\gamma(P)\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$;
hence, if $P=\sum_{g\in G}a_{g}g$, with $a_{g}\in\operatorname{\mathbb{C}}$
for all $g\in G$, we have

$P^{(n)}(e_{x})=\sum_{g\in G}a_{g}e_{g(x)},$

for every $x\in[n]^{k}$. The set
$\\{P^{(n)}:P\in\operatorname{\mathbb{C}}[S_{n}]\\}$ coincides with the set of
endomorphisms of $V^{\otimes k}$ commuting with the action of
$\operatorname{\mathrm{GL}}(V)$ on $V^{\otimes k}$ defined by setting
$h(v_{1}\otimes\ldots\otimes v_{k})=h(v_{1})\otimes\ldots\otimes h(v_{k})$,
for all $h\in\operatorname{\mathrm{GL}}(V)$, $v_{1},\ldots,v_{k}\in V$ (see
e.g. [7, Theorem 8.2.8]).

For a group $G$, let $\mathbf{1}_{G}:G\rightarrow\\{1\\}$ be its trivial
character. If $\chi_{1}:G\rightarrow\operatorname{\mathbb{C}}$ and
$\chi_{2}:G\rightarrow\operatorname{\mathbb{C}}$ are characters, their scalar
product is defined by $\langle\chi_{1},\chi_{2}\rangle:=\sum_{g\in
G}\chi_{1}(g)\chi_{2}(g^{-1})$. If $\chi$ is a simple character of a subgroup
$G\subseteq S_{k}$, then the element

$P_{\chi}:=\frac{\chi(e)}{|G|}\sum_{g\in
G}\chi(g^{-1})g\in\operatorname{\mathbb{C}}[S_{k}]$

is an idempotent corresponding to the induced representation
$\mathrm{Ind}^{S_{k}}_{G}(\chi)$ of $S_{k}$. For such idempotents, we write
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)$ instead of
$\operatorname{\mathrm{Gr}}_{P_{\chi}^{(n)}}(k,n)$. 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 $\chi:=\mathbf{1}_{G}$. In this case
$P_{\chi}^{(n)}=\operatorname{\mathrm{Id}}_{V^{\otimes k}}$ and
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)=\operatorname{\mathrm{Seg}}(k,n)$.

Grassmannians: if $G=S_{k}$ and $\chi$ is its alternating character, then

$P_{\chi}=\frac{1}{k!}\sum_{\sigma\in S_{k}}\mathrm{sgn}(\sigma)\sigma,$

where $\mathrm{sgn}(\sigma)$ denotes the sign of the permutation $\sigma$.
Hence, as vector spaces,
$\operatorname{\mathrm{Im}}\left(P_{\chi}^{(n)}\right)\simeq\bigwedge^{k}V$
and, for $k\leqslant n$,
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)=\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$
is the complex Grassmannian
$\operatorname{\mathrm{Gr}}_{\operatorname{\mathbb{C}}}(k,n)$; for $k>n$ it is
clear that $\operatorname{\mathrm{Gr}}_{\chi}(k,n)=\varnothing$.

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 $\chi:=\mathbf{1}_{S_{k}}$, then

$P_{\chi}=\frac{1}{k!}\sum_{\sigma\in S_{k}}\sigma.$

Hence, as vector spaces,
$\operatorname{\mathrm{Im}}\left(P_{\chi}^{(n)}\right)\simeq\mathrm{Sym}^{k}V$
and
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)=\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$
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 $h\in\operatorname{\mathbb{N}}$,
$a\in\operatorname{\mathbb{N}}^{h}$ and $k:=a(h)$, where
$a(j):=\sum_{i=1}^{j}a_{i}$, for all $j\in[h]$. The symmetric group $S_{k}$ is
generated by the simple transpositions $\\{s_{1},\ldots,s_{k-1}\\}$; for
$i,j\in[k-1]$, $i\leqslant j$, define the parabolic subgroup

$S_{[i,j]}:=\left\\{\begin{array}[]{ll}\langle
s_{i},s_{i+1},\ldots,s_{j-1}\rangle,&\hbox{ if $i<j$;}\\\ \\{e\\},&\hbox{ if
$i=j$.}\end{array}\right.$

We define the parabolic subgroup $G_{a}\subseteq S_{k}$ by

$G_{a}:=S_{[1,a(1)]}\times S_{[a(1)+1,a(2)]}\times\ldots\times
S_{[a(h-1)+1,a(h)]}.$

Hence $P_{\textbf{1}_{G_{a}}}=\frac{1}{a_{1}!\cdots a_{h}!}\sum_{g\in
G_{a}}g\in\operatorname{\mathbb{C}}[S_{k}]$ is the idempotent corresponding to
the Young module $M^{a}$ (see e.g. [7, Section 3.6.2]). Since, for $k\geqslant
1$, the Chow variety $G(1,k,2)$ is isomorphic to
$\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{k+1})$ (because any
homogeneous polynomial of positive degree in two variables factorizes as
product of degree one polynomials), we obtain

$\operatorname{\mathrm{Gr}}_{\textbf{1}_{G_{a}}}(k,2)\simeq\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{a_{1}+1})\times\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{a_{2}+1})\times\ldots\times\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{a_{h}+1}).$
(1)

For a simple character $\chi$ of $G\subseteq S_{k}$, define $V^{\otimes
k}_{\chi}:=\operatorname{\mathrm{Im}}\left({P^{(n)}_{\chi}}\right)$. We denote
by $\mathrm{Res}^{G}_{H}(\chi)$ the restriction to a group $H\subseteq G$ of a
character $\chi$ of $G$. Given an action of a group $G$ on a set $X$, let
$G_{x}$ be the _isotropy group_ of the element $x\in X$, i.e. $G_{x}:=\\{g\in
G:g(x)=x\\}$.

A set of generators and the dimension of the vector space $V^{\otimes
k}_{\chi}$ are known, see [24, Eq. 6.13 and 6.23].

######  Theorem 3.9.

Let $G\subseteq S_{k}$ be a subgroup and $\chi$ a simple character of $G$.
Then $P^{(n)}_{\chi}(e_{x})=0$ if and only if
$\left\langle\mathrm{Res}^{G}_{G_{x}}(\chi),\mathbf{1}_{G_{x}}\right\rangle=0$,
for every $x\in[n]^{k}$, and

$\dim\left(V^{\otimes k}_{\chi}\right)=\frac{\chi(e)}{|G|}\sum\limits_{g\in
G}\chi(g)n^{c(g)},$

where $c(g)$ is the number of cycles of the permutation $g\in S_{k}$.

In particular, by Theorem 3.9, we have that

$V^{\otimes
k}_{\chi}=\operatorname{span}_{\operatorname{\mathbb{C}}}\left\\{P^{(n)}_{\chi}(e_{x}):\left\langle\mathrm{Res}^{G}_{G_{x}}(\chi),\mathbf{1}_{G_{x}}\right\rangle\neq
0\right\\}.$

For a more explicit description of the spanning set in Theorem 3.9 when
$G=S_{k}$, see [24, Theorem 6.37]. In general, this set is not a basis, as
shown in the following example. Moreover, this is an example of immanant
variety where
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)\neq\operatorname{\mathrm{Im}}\left(\widehat{P}_{\chi}^{(n)}\right)$.

######  Example 3.10.

Let $n=2$, $k=3$, $G=S_{3}$ and $\chi$ the two-dimensional simple character of
$S_{3}$ corresponding to the partition $(2,1)$. Then
$P_{\chi}=\frac{1}{3}\left(2e-st-ts\right)$, where $s=213$ and $t=132$ are the
simple transpositions (we write permutations in one-line notation). By Theorem
3.9, $\dim\left(V^{\otimes k}_{\chi}\right)=4$. We have that
$P^{(2)}_{\chi}(e_{(1,1,1)})=P^{(2)}_{\chi}(e_{(2,2,2)})=0$ and

  * •

$P^{(2)}_{\chi}(e_{(1,1,2)})=\frac{1}{3}\left(2e_{(1,1,2)}-e_{(1,2,1)}-e_{(2,1,1)}\right)$,

  * •

$P^{(2)}_{\chi}(e_{(1,2,1)})=\frac{1}{3}\left(2e_{(1,2,1)}-e_{(2,1,1)}-e_{(1,1,2)}\right)$,

  * •

$P^{(2)}_{\chi}(e_{(2,1,1)})=\frac{1}{3}\left(2e_{(2,1,1)}-e_{(1,1,2)}-e_{(1,2,1)}\right)$.

Hence
$P^{(2)}_{\chi}(e_{(1,2,1)})=-P^{(2)}_{\chi}(e_{(1,1,2)})-P^{(3,2)}_{\chi}(e_{(2,1,1)})$
(similarly for $P^{(2)}_{\chi}\left(e_{(2,1,2)}\right)$) and a basis for
$V^{\otimes 3}_{\chi}$ is

$\left\\{P^{(2)}_{\chi}(e_{(1,1,2)}),P^{(2)}_{\chi}(e_{(2,1,1)}),P^{(2)}_{\chi}(e_{(2,2,1)}),P^{(2)}_{\chi}(e_{(1,2,2)})\right\\}.$

Parametric equations for
$\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$ are

$\left\\{\begin{array}[]{ll}x_{111}=0\\\
x_{112}=2a_{1}b_{1}c_{2}-a_{2}b_{1}c_{1}-a_{1}b_{2}c_{1}\\\
x_{121}=2a_{1}b_{2}c_{1}-a_{1}b_{1}c_{2}-a_{2}b_{1}c_{1}\\\
x_{122}=2a_{1}b_{2}c_{2}-a_{2}b_{1}c_{2}-a_{2}b_{2}c_{1}\\\
x_{211}=2a_{2}b_{1}c_{1}-a_{1}b_{2}c_{1}-a_{1}b_{1}c_{2}\\\
x_{212}=2a_{2}b_{1}c_{2}-a_{2}b_{2}c_{1}-a_{1}b_{2}c_{2}\\\
x_{221}=2a_{2}b_{2}c_{1}-a_{1}b_{2}c_{2}-a_{2}b_{1}c_{2}\\\
x_{222}=0\end{array}\right.$

where
$(a_{1},a_{2}),(b_{1},b_{2}),(c_{1},c_{2})\in\operatorname{\mathbb{C}}^{2}\setminus\\{(0,0)\\}$.
Using Gröbner basis method for implicitization, we find the following
Cartesian equations for $\operatorname{\mathrm{Gr}}_{\chi}(3,2)$:

$\left\\{\begin{array}[]{ll}x_{112}+x_{121}+x_{211}=0\\\
x_{221}+x_{212}+x_{122}=0\\\ x_{111}=0\\\ x_{222}=0\end{array}\right.$

These equations are the ones for $\operatorname{\mathbb{P}}(V^{\otimes
3}_{\chi})$ in $\operatorname{\mathbb{P}}(V^{\otimes 3})$; hence
$\operatorname{\mathrm{Gr}}_{\chi}(3,2)=\operatorname{\mathbb{P}}(V^{\otimes
3}_{\chi})\simeq\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{4})$. It
is not difficult to see that
$\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)\subsetneq\operatorname{\mathrm{Gr}}_{\chi}(3,2)$;
in fact it can be checked by hand that

$\left[P^{(2)}_{\chi}(e_{(1,1,2)})-P^{(2)}_{\chi}(e_{(2,1,1)})\right]\in\operatorname{\mathbb{P}}(V^{\otimes
3}_{\chi})\setminus\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right).$

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
$G\subseteq S_{k}$ and a simple character of $G$. Our definition extends to
submatrices the notion of generalized matrix function (see [24, Chapter 7]).

Let $k,m,n\in\operatorname{\mathbb{N}}$; recall that $V$ is an $n$-dimensional
complex vector space. If $W$ is an $m$-dimensional complex vector space and
$f\in\operatorname{\mathrm{Hom}}(V,W)$, the element $f^{\otimes
k}\in\operatorname{\mathrm{Hom}}(V^{\otimes k},W^{\otimes k})$ is defined by
setting

$f^{\otimes k}(e_{x})=f(e_{x_{1}})\otimes\ldots\otimes f(e_{x_{k}}),$

for all $x\in[n]^{k}$.

######  Remark 3.11.

When $W=V$, we have that $(f\circ g)^{\otimes k}=f^{\otimes k}\circ g^{\otimes
k}$, for all $f,g\in\operatorname{\mathrm{End}}(V)$, and
$(\operatorname{\mathrm{Id}}_{V})^{\otimes
k}=\operatorname{\mathrm{Id}}_{V^{\otimes k}}$, i.e. the function $f\mapsto
f^{\otimes k}$ defines a monoid morphism
$\operatorname{\mathrm{End}}(V)\rightarrow\operatorname{\mathrm{End}}(V^{\otimes
k})$.

Given a matrix
$M\in\operatorname{\mathrm{Mat}}_{m,n}(\operatorname{\mathbb{C}})$, the matrix
$M^{\otimes
k}\in\operatorname{\mathrm{Mat}}_{m^{k},n^{k}}(\operatorname{\mathbb{C}})$ is
defined by setting

$\left(M^{\otimes k}\right)_{x,y}=\prod_{i=1}^{k}M_{x_{i},y_{i}},$

for all $x\in[m]^{k}$, $y\in[n]^{k}$, i.e. if $M$ is the matrix associated to
$f\in\operatorname{\mathrm{Hom}}(V,W)$ with respect to bases of $V$ and $W$,
the matrix associated to $f^{\otimes k}$ is $M^{\otimes k}$. In the following
definition we extend the notion of immanant of a square matrix.

######  Definition 3.12.

Let $x\in[m]^{k}$, $y\in[n]^{k}$ and $\chi$ a simple character of a group
$G\subseteq S_{k}$. The $\chi_{x,y}$-_immanant_ of a matrix
$M\in\operatorname{\mathrm{Mat}}_{m,n}(\operatorname{\mathbb{C}})$ is defined
by

$\chi_{x,y}(M):=\sum\limits_{g\in G}\chi(g)\left(M^{\otimes
k}\right)_{g(x),y}.$

Let $M\in\operatorname{\mathrm{Mat}}_{n,n}(\operatorname{\mathbb{C}})$, $k=n$
and $x=y=(1,\ldots,n)=:\vec{n}$. In this setting, for $G=S_{n}$, we recover
some well-known numbers associated to $M$.

  * •

If $\chi$ is the alternating character of $S_{n}$, then
$\chi_{\vec{n},\vec{n}}(M)$ is the _determinant_ of the matrix $M$.

  * •

If $\chi:=\mathbf{1}_{S_{n}}$, then $\chi_{\vec{n},\vec{n}}(M)$ is the
_permanent_ of the matrix $M$.

  * •

If $\chi$ is any simple character of $S_{n}$, then $\chi_{\vec{n},\vec{n}}(M)$
is the so-called _immanant_ of the matrix $M$, see for instance [29] and
references therein.

######  Example 3.13.

Let $M\in\operatorname{\mathrm{Mat}}_{2,3}(\operatorname{\mathbb{C}})$ be the
generic matrix

$M=\begin{pmatrix}a_{11}&a_{12}&a_{13}\\\ a_{21}&a_{22}&a_{23}\end{pmatrix}.$

Then, by ordering $[2]^{2}$ and $[3]^{2}$ lexicographically, the matrix
$M^{\otimes 2}$ is

$\begin{pmatrix}a_{11}^{2}&a_{11}a_{12}&a_{11}a_{13}&a_{11}a_{12}&a_{12}^{2}&a_{12}a_{13}&a_{11}a_{13}&a_{12}a_{13}&a_{13}^{2}\\\
a_{11}a_{21}&a_{11}a_{22}&a_{11}a_{23}&a_{12}a_{21}&a_{12}a_{22}&a_{12}a_{23}&a_{13}a_{21}&a_{13}a_{22}&a_{13}a_{23}\\\
a_{21}a_{11}&a_{21}a_{12}&a_{21}a_{13}&a_{22}a_{11}&a_{22}a_{12}&a_{22}a_{13}&a_{23}a_{11}&a_{23}a_{12}&a_{23}a_{13}\\\
a_{21}^{2}&a_{21}a_{22}&a_{21}a_{23}&a_{22}a_{21}&a_{22}^{2}&a_{22}a_{23}&a_{23}a_{21}&a_{23}a_{22}&a_{23}^{2}\end{pmatrix}.$

If $\chi$ is the trivial character of $S_{2}$, we have
$\chi_{(2,2),(2,3)}(M)=2a_{22}a_{23}$. If $\chi$ is the alternating character
of $S_{2}$, we have $\chi_{(1,2),(1,3)}(M)=a_{11}a_{23}-a_{21}a_{13}$ and
$\chi_{(2,2),(2,3)}(M)=0$.

Let $P\in\operatorname{\mathbb{C}}[S_{k}]$; it is clear that

$f^{\otimes k}\circ P^{(n)}=P^{(m)}\circ f^{\otimes k},$

i.e. $f^{\otimes
k}\in\operatorname{\mathrm{Hom}}_{\operatorname{\mathbb{C}}[S_{k}]}\left(V^{\otimes
k},W^{\otimes k}\right)$, for all $f\in\operatorname{\mathrm{Hom}}(V,W)$.

The $\chi_{x,y}$-immanant of a matrix $M$ is related to the $(x,y)$-entry of
the matrix $M^{\otimes k}P^{(n)}_{\chi}$, as the next proposition asserts.

######  Proposition 3.14.

Let $x\in[m]^{k}$, $y\in[n]^{k}$, $\chi$ a simple character of a group
$G\subseteq S_{k}$ and
$M\in\operatorname{\mathrm{Mat}}_{m,n}(\operatorname{\mathbb{C}})$. Then

$\left(M^{\otimes
k}P^{(n)}_{\chi}\right)_{x,y}=\frac{\chi(e)}{|G|}\chi_{x,y}(M).$

###### Proof.

Let $y\in[n]^{k}$ and $\\{w_{i}:i\in[m]\\}$ be a basis of the $m$-dimensional
vector space $W$. We have that

$\displaystyle M^{\otimes k}P^{(n)}_{\chi}e_{y}$ $\displaystyle=$
$\displaystyle P^{(m)}_{\chi}M^{\otimes k}e_{y}$ $\displaystyle=$
$\displaystyle\frac{\chi(e)}{|G|}\sum\limits_{g\in
G}\chi(g^{-1})\sum\limits_{x\in[m]^{k}}\left(M^{\otimes
k}\right)_{x,y}w_{g(x)}$ $\displaystyle=$
$\displaystyle\frac{\chi(e)}{|G|}\sum\limits_{g\in
G}\chi(g^{-1})\sum\limits_{x\in[m]^{k}}\left(M^{\otimes
k}\right)_{g^{-1}(x),y}w_{x}$ $\displaystyle=$
$\displaystyle\frac{\chi(e)}{|G|}\sum\limits_{x\in[m]^{k}}\left(\sum\limits_{g\in
G}\chi(g)\left(M^{\otimes k}\right)_{g(x),y}\right)w_{x}.$

∎

######  Remark 3.15.

For $P\in\operatorname{\mathbb{C}}[S_{k}]$, we can define, by restriction, an
element $f^{\otimes
k}_{P}\in\operatorname{\mathrm{Hom}}\left(\operatorname{\mathrm{Im}}(P^{(n)}),\operatorname{\mathrm{Im}}(P^{(m)})\right)$;
if $P=P_{\chi}$ for some simple character $\chi$ of a group $G\subseteq
S_{k}$, then $f^{\otimes
k}_{P}\in\operatorname{\mathrm{Hom}}_{\operatorname{\mathbb{C}}[G]}\left(\operatorname{\mathrm{Im}}(P^{(n)}),\operatorname{\mathrm{Im}}(P^{(m)})\right)$,
since $gP_{\chi}=P_{\chi}g$, for all $g\in G$, and then
$\operatorname{\mathrm{Im}}(P^{(n)})$, $\operatorname{\mathrm{Im}}(P^{(m)})$
are $\operatorname{\mathbb{C}}[G]$-modules. See [22] for a more extended
treatment on such induced morphisms.

Now we provide parametric equations for the set
$\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$ in terms of
immanants. Let
$A\in\operatorname{\mathrm{Mat}}_{n,k}(\operatorname{\mathbb{C}})$ defined by

$A:=\left(\begin{array}[]{cccc}a_{11}&a_{12}&\cdots&a_{1k}\\\
a_{21}&a_{22}&\cdots&a_{2k}\\\ \vdots&\vdots&\cdots&\vdots\\\
a_{n1}&a_{n2}&\cdots&a_{nk}\\\ \end{array}\right)$

where none of the columns is the zero vector.

######  Theorem 3.16.

Let $k,n\in\operatorname{\mathbb{N}}$ and $\chi$ be a simple character of a
group $G\subseteq S_{k}$. Then the set
$\operatorname{\mathrm{Im}}(\widehat{P}^{(n)}_{\chi})\subseteq\operatorname{\mathbb{P}}\left(V^{\otimes
k}\right)$ is described by the parametric equations

$\left\\{x_{z}={\chi}_{z,(1,\ldots,k)}(A):z\in[n]^{k}\right\\},$

where $A$ is the generic matrix defined above.

###### Proof.

Let $v_{i}=a_{1i}e_{1}+\ldots+a_{ni}e_{n}$, for all $i\in[k]$, and let $W$ be
a vector space with basis $\\{\tilde{e}_{1},\ldots,\tilde{e}_{k}\\}$. Then
$A\in\operatorname{\mathrm{Hom}}(W,V)$ and, by Proposition 3.14,

$\displaystyle P^{(n)}_{\chi}(v_{1}\otimes\ldots\otimes v_{k})$
$\displaystyle=$ $\displaystyle
P^{(n)}_{\chi}\left((A\tilde{e}_{1})\otimes\ldots\otimes(A\tilde{e}_{k})\right)$
$\displaystyle=$ $\displaystyle P^{(n)}_{\chi}A^{\otimes
k}\tilde{e}_{(1,...,k)}$ $\displaystyle=$ $\displaystyle A^{\otimes
k}P^{(k)}_{\chi}\tilde{e}_{(1,...,k)}$ $\displaystyle=$
$\displaystyle\frac{\chi(e)}{|G|}\sum\limits_{x\in[n]^{k}}\chi_{x,(1,\ldots,k)}(A)\tilde{e}_{x}.$

∎

The statement of Theorem 3.16 leads us to the following definition.

######  Definition 3.17.

If $\chi$ is a simple character of a group $G\subseteq S_{k}$, we call
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)$ an _immanant variety_.

We observe that, if two subgroups $G\subseteq S_{k}$ and $H\subseteq S_{k}$
are conjugated in $S_{k}$, i.e. $H=\sigma G\sigma^{-1}$ for some $\sigma\in
S_{k}$, and $\chi$ is a character of $G$, then, as projective varieties,
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)\simeq\operatorname{\mathrm{Gr}}_{\chi^{\sigma}}(k,n)$,
where $\chi^{\sigma}(h):=\chi(\sigma^{-1}h\sigma)$, for all $h\in H$. In fact
$P_{\chi^{\sigma}}=\sigma P_{\chi}\sigma^{-1}$. Let
$f:=\gamma(P_{\chi}\sigma^{-1})\in\operatorname{\mathrm{End}}\left(V^{\otimes
k}\right)$, where
$\gamma:\operatorname{\mathbb{C}}[S_{k}]\rightarrow\operatorname{\mathrm{End}}\left(V^{\otimes
k}\right)$ is the algebra morphism introduced above. Then
$\operatorname{\mathrm{Im}}(\,\widehat{f}\,)=\operatorname{\mathrm{Im}}(\widehat{P}^{(n)}_{\chi})$
and

$\operatorname{\mathrm{Gr}}_{\chi^{\sigma}}(k,n)=\operatorname{\mathrm{Gr}}_{\gamma(\sigma)\circ
f}(k,n)\simeq\operatorname{\mathrm{Gr}}_{f}(k,n)=\operatorname{\mathrm{Gr}}_{\chi}(k,n),$

by Proposition 3.4. In the following example we see that $G\simeq H$ and
$\chi_{G}\simeq\chi_{H}$ do not imply
$\operatorname{\mathrm{Gr}}_{\chi_{G}}(k,n)\simeq\operatorname{\mathrm{Gr}}_{\chi_{H}}(k,n)$
as projective varieties.

######  Example 3.18.

Let $G:=\\{e,2134\\}\subseteq S_{4}$ and $H:=\\{e,4321\\}\subseteq S_{4}$.
Then
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(4,2)\simeq\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{3})\times\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{2})\times\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{2})$,
by (1), because $G=G_{(2,1,1)}$. Hence it is a smooth four-dimensional
projective variety. On the other hand, we checked by using Macaulay2 [17] that
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{H}}(4,2)$ is a singular four-
dimensional projective varieties. Then, as projective varieties, they are not
isomorphic.

##  4 One-dimensional characters and $\chi$-matroids

In this section we restrict our attention to one-dimensional characters of a
subgroup $G\subseteq S_{k}$. This includes, for example, the trivial character
of $G$ and all the simple characters of an abelian group.

For $x\in[n]^{k}$ we let $O_{x}:=\\{g(x):g\in G\\}\subseteq[n]^{k}$ and
$\preccurlyeq_{\operatorname{\mathrm{lex}}}$ the lexicographic order on
$O_{x}$. Then we define

$\overline{x}:=\min\left(O_{x},\preccurlyeq_{\operatorname{\mathrm{lex}}}\right).$
(2)

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 $G\subseteq S_{k}$ be a subgroup and $\chi$ a one-dimensional character of
$G$, i.e. a group morphism
$\chi:G\rightarrow\operatorname{\mathbb{C}}\setminus\\{0\\}$. Then

$\left\\{P^{(n)}_{\chi}(e_{\overline{x}}):x\in[n]^{k},\,G_{x}\subseteq\ker(\chi)\right\\}$

is a basis of $V^{\otimes k}_{\chi}$.

###### Proof.

By Theorem 3.9 we have that $P^{(n)}_{\chi}(e_{x})=0$ if and only if
$\sum\limits_{g\in G_{x}}\chi(g)=0$ and this is equivalent to
$G_{x}\not\subseteq\ker(\chi)$. In fact, it is straightforward to see that
$\sum\limits_{g\in G_{x}}\chi(g)=0$ implies $G_{x}\not\subseteq\ker(\chi)$. On
the other hand, let $h\in G_{x}\setminus\ker(\chi)$; therefore
$\sum\limits_{g\in G_{x}}\chi(g)=\sum\limits_{g\in
G_{x}}\chi(hg)=\chi(h)\sum\limits_{g\in G_{x}}\chi(g)$ and we find that
$(1-\chi(h))\sum\limits_{g\in G_{x}}\chi(g)=0$. This implies
$\sum\limits_{g\in G_{x}}\chi(g)=0$.

Let $x,y\in[n]^{k}$ with $G_{x}\subseteq\ker(\chi)$ and
$G_{y}\subseteq\ker(\chi)$. Moreover assume $y=g(x)$ for some $g\in G$. Notice
that

$P^{(n)}_{\chi}(e_{x})=\frac{1}{|G/G_{x}|}\sum\limits_{c\in
G/G_{x}}\chi(\tilde{c}^{-1})e_{\tilde{c}(x)},$

for all $x\in[n]^{k}$, where $\tilde{c}$ is any representative of the class
$c\in G/G_{x}$. By using this formula, it can be easily shown that
$P^{(n)}_{\chi}(e_{y})=\chi(g)P^{(n)}_{\chi}(e_{x})$. It follows that it is
sufficient to choose an element for each orbit $O_{x}$, for instance
$\overline{x}$. Since
$\overline{x}_{1},\overline{x}_{2},...,\overline{x}_{h}\in[n]^{k}$ are in $h$
distinct orbits then clearly $P^{(n)}_{\chi}(e_{\overline{x}_{1}})$,
$P^{(n)}_{\chi}(e_{\overline{x}_{2}})$, $\ldots$,
$P^{(n)}_{\chi}(e_{\overline{x}_{h}})$ are linearly independent. This
concludes the proof. ∎

We define

$B_{\chi}(k,n):=\left\\{\overline{x}:x\in[n]^{k},\,G_{x}\subseteq\ker{\chi}\right\\},$

ordered by setting

$\mbox{$x\preccurlyeq y$ if and only if $x\leqslant g(y)$},$

for some $g\in G$, for all $x,y\in B_{\chi}(k,n)$, where $\leqslant$ is
componentwise. We need to prove that this is really a partial order.

######  Proposition 4.2.

The relation $\preccurlyeq$ on $B_{\chi}(k,n)$ is a partial order.

###### Proof.

We claim that $x\leqslant g(x)$ if and only if $g\in G_{x}$. One implication
is obvious. Let $x\leqslant g(x)$ for some $g\in G$. Let $\rho$ be the rank
function of the poset $([n]^{k},\leqslant)$. Then $\rho(x)=\rho(g(x))$. This
implies that $g(x)=x$.

  1. 1.

reflexivity: clearly $x\leqslant e(x)=x$ and then $x\preccurlyeq x$, for all
$x\in B_{\chi}(k,n)$.

  2. 2.

symmetry: let $x\preccurlyeq y$ and $y\preccurlyeq x$. Then $x\leqslant g(y)$
and $y\leqslant h(x)$, for some $g,h\in G$. Therefore $x\leqslant
g(y)\leqslant gh(x)$. Hence $x\leqslant gh(x)$ and by our claim we conclude
that $gh(x)=x$. Then $x=g(y)$, i.e. $y\in O_{x}$ and $x=y$.

  3. 3.

transitivity: let $x\preccurlyeq y$ and $y\preccurlyeq z$; then $x\leqslant
g(y)$ and $y\leqslant h(z)$ for some $g,h\in G$. This implies $x\leqslant
g(y)\leqslant gh(z)$, i.e. $x\preccurlyeq z$.

∎

For example, if $G=\\{e\\}$ is the trivial group, then
$B_{\chi}(k,n)=\left([n]^{k},\leqslant\right)$. When $\chi$ is the alternating
character of $S_{k}$, the poset $B_{\chi}(k,n)$ is isomorphic to the poset
$S_{n}^{(k)}$ of Grassmannian permutations with the Bruhat order (see [3,
Proposition 4.9]).

######  Example 4.3.

[Lyndon words] Let $C_{k}:=\\{e,k12\ldots k-1,\ldots,23\ldots 1\\}\subseteq
S_{k}$ be a cyclic group of order $k$, $\sigma:=k12\ldots k-1\in C_{k}$ and
$\chi:C_{k}\rightarrow\operatorname{\mathbb{C}}\setminus\\{0\\}$ the character
defined by $\chi(\sigma^{h})=\exp\left(\frac{2\pi hi}{k}\right)$, for all
$h\in[k]$. Since $\ker(\chi)=\\{e\\}$, we have that $B_{\chi}(k,n)$ is the
poset of _Lyndon words_ of length $k$ over the alphabet $[n]$. The poset
$B_{\chi}(k,n)$ has maximum $(n-1,n,\ldots,n)$ and minimum $(1,\ldots,1,2)$.
The cardinality of $B_{\chi}(k,n)$ is given by Witt’s formula:

$|B_{\chi}(k,n)|=\frac{1}{k}\sum\limits_{d\mid k}\mu(d)n^{k/d},$

where $\mu$ is the Möbius function. For more information on Lyndon words, see
e.g. [28, Exercise 7.89]. The following is the Hasse diagram of
$B_{\chi}(6,2)$.

$(1,2,2,2,2,2)$$(1,1,2,2,2,2)$$(1,2,1,2,2,2)$$(1,1,2,2,1,2)$$(1,1,1,2,2,2)$$(1,1,2,1,2,2)$$(1,1,1,1,2,2)$$(1,1,1,2,1,2)$$(1,1,1,1,1,2)$

Now we extend the notion of rank $k$ matroids to all one-dimensional
characters $\chi$ of a group $G\subseteq S_{k}$. Let $\sigma\in S_{n}$; then
$\sigma$ acts on $[n]^{k}$ by letting

$\sigma^{*}(x):=(\sigma(x_{1}),\ldots,\sigma(x_{k})),$

for all $x\in[n]^{k}$. Hence we have an action of $S_{n}$ on $V^{\otimes k}$,
commuting with the previously defined action of $S_{k}$ on $V^{\otimes k}$.

######  Definition 4.4.

Let $X\subseteq B_{\chi}(k,n)$. We say that $X$ is a $\chi$-matroid if, for
every $\sigma\in S_{n}$, the induced subposet
$\left\\{\overline{\sigma^{*}(x)}:x\in X\right\\}\subseteq B_{\chi}(k,n)$ has
a unique maximum, where for $x\in[n]^{k}$, $\overline{x}$ 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 $\chi$ is the alternating character of
$G=S_{k}$, we recover the set of bases of a rank $k$ matroid on the ground set
$[n]$.

For a one-dimensional character $\chi$ of $G\subseteq S_{k}$ and
$[v]\in\operatorname{\mathbb{P}}(V^{\otimes k}_{\chi})$, we define a set
$\operatorname{\mathrm{supp}}_{\chi}[v]\subseteq B_{\chi}(k,n)$ by letting

$\operatorname{\mathrm{supp}}_{\chi}[v]:=\left\\{x\in B_{\chi}(k,n):a_{x}\neq
0\right\\},$

where $v=\sum\limits_{x\in B_{\chi}(k,n)}a_{x}P^{(n)}_{\chi}(e_{x})$.

We now introduce the _maximality property_ for
$\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$, whenever
$\chi$ is a one-dimensional character of a subgroup $G\subseteq S_{k}$.

######  Definition 4.6.

Let $\chi$ be a one-dimensional character of a subgroup $G\subseteq S_{k}$. We
say that $\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$ has
the _maximality property_ if the induced subposet
$\operatorname{\mathrm{supp}}_{\chi}[v]$ is a $\chi$-matroid, for all
$[v]\in\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$.

In the following example we see a case in which the maximality property does
not hold.

######  Example 4.7.

Let $n=3$, $k=4$ and $G=\\{e,g\\}=\\{e,3412\\}\subseteq S_{4}$ and
$\chi:G\rightarrow\operatorname{\mathbb{C}}$ defined by setting
$\chi(3412)=-1$. Then the action of the group $G$ on $V^{\otimes 4}$ is given
by
$g\left(e_{(i_{1},i_{2},i_{3},i_{4})}\right)=e_{(i_{3},i_{4},i_{1},i_{2})}$,
for all $i_{1},i_{2},i_{3},i_{4}\in[3]$. We have that
$P^{(3)}_{\chi}(e_{(3,3,3,3)})=0$ and that the poset $B_{\chi}(4,3)$ has
maxima $(2,3,3,3)$ and $(3,2,3,3)$. Let
$v:=P^{(3)}_{\chi}\left((e_{2}+e_{3})\otimes(e_{2}+e_{3})\otimes e_{3}\otimes
e_{3}\right)$; therefore

$\displaystyle v$ $\displaystyle=$ $\displaystyle
P^{(3)}_{\chi}(e_{(2,2,3,3)})+P^{(3)}_{\chi}(e_{(2,3,3,3)})+P^{(3)}_{\chi}(e_{(3,2,3,3)}).$

Hence
$\operatorname{\mathrm{supp}}_{\chi}[v]=\\{(2,2,3,3),(2,3,3,3),(3,2,3,3)\\}$
is not a $\chi$-matroid because this poset has two maximal elements.

Several important varieties have the maximality property. For example, the
Grassmannian $\operatorname{\mathrm{Gr}}_{\operatorname{\mathbb{C}}}(k,n)$ has
this property. In fact, the elements of
$\operatorname{\mathrm{Gr}}_{\operatorname{\mathbb{C}}}(k,n)$ 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.

######  Proposition 4.8.

The set $\operatorname{\mathrm{Seg}}(k,n)$ has the maximality property.

###### Proof.

For $k\in\operatorname{\mathbb{N}}$, let $e(k)$ be the identity in $S_{k}$.
The statement is equivalent to say that
$\operatorname{\mathrm{supp}}_{\mathbf{1}_{\\{e(k)\\}}}[v_{1}\otimes\ldots\otimes
v_{k}]$ has a unique maximum for all $v_{1},\ldots,v_{k}\in
V\setminus\\{0\\}$. Let $v:=v_{1}\otimes\ldots\otimes v_{k}$; hence

$\max(\operatorname{\mathrm{supp}}_{\mathbf{1}_{\\{e(k)\\}}}[v])=\left(\max(\operatorname{\mathrm{supp}}_{\mathbf{1}_{\\{e(1)\\}}}[v_{1}]),\ldots,\max(\operatorname{\mathrm{supp}}_{\mathbf{1}_{\\{e(1)\\}}}[v_{k}])\right)$

and this concludes the proof. ∎

More in general, when $\chi$ is a trivial character, the set
$\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$ 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 $X\subseteq B_{\chi}(k,n)$ is _representable_ over
$\operatorname{\mathbb{C}}$ if
$X\in\left\\{\operatorname{\mathrm{supp}}_{\chi}[v]:[v]\in\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)\right\\}$.

Notice that $\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$
has the maximality property if and only all representable subsets $X\subseteq
B_{\chi}(k,n)$ are $\chi$-matroids. Differently to the Grassmannian case,
there exist representable subsets in $B_{\chi}(k,n)$ which are not
$\chi$-matroids, see Example 4.7.

######  Remark 4.10.

Let $m:=\dim\left(V_{\chi}^{\otimes k}\right)$. If
$\operatorname{\mathrm{Gr}}_{\chi}(k,n)=\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\chi}\right)$
and the variety $\operatorname{\mathrm{Gr}}_{\chi}(k,n)$ has the maximality
property, then the action on $\operatorname{\mathbb{P}}\left(V_{\chi}^{\otimes
k}\right)$ 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 $\operatorname{\mathrm{Gr}}_{\chi}(k,n)$ whose
strata are in bijection with representable $\chi$-matroids. If $\chi$ 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].

##  5 The combinatorics of $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$

In this section we consider the trivial character $\mathbf{1}_{G}$ of a
subgroup $G\subseteq S_{k}$. In this case
$B_{\mathbf{1}_{G}}(k,n)=\\{\overline{x}:x\in[n]^{k}\\}$ because the condition
$G_{x}\subseteq\ker(\mathbf{1}_{G})=G$ (see Theorem 4.1) is trivially
satisfied. Moreover we have that
$\operatorname{\mathbb{P}}\left(\ker\left(P^{(n)}_{\mathbf{1}_{G}}\right)\right)\cap\operatorname{\mathrm{Seg}}(k,n)=\varnothing$;
in fact we can state the following more general result.

######  Proposition 5.1.

Let $P\in\operatorname{\mathbb{C}}[S_{k}]$ be such that $P^{(n)}(e_{x})\neq 0$
for all $x\in[n]^{k}$. Then
$\operatorname{\mathrm{Gr}}_{P^{(n)}}(k,n)=\operatorname{\mathrm{Im}}(\widehat{P}^{(n)})$.

###### Proof.

Let $v:=v_{1}\otimes\ldots\otimes v_{k}\in V^{\otimes k}$. Hence
$\max\left(\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}[v]\right)=\\{x\\}\subseteq[n]^{k}$,
by the maximality property of $\operatorname{\mathrm{Seg}}(k,n)$. Let $\rho$
be the rank function of $[n]^{k}$; then $\rho(y)<\rho(x)=\rho(z)$, for all
$y\in\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}[v]\setminus\\{x\\}$ and for
all
$z\in\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}\left[P^{(n)}(e_{x})\right]$,
because the action of $S_{k}$ on $[n]^{k}$ preserves its rank. This implies
$P^{(n)}(v)\neq 0$; hence
$\operatorname{\mathrm{Seg}}(k,n)\cap\operatorname{\mathbb{P}}\left(\ker\left(P^{(n)}\right)\right)=\varnothing$
and the result follows by Proposition 3.3. ∎

######  Corollary 5.2.

Let $G\subseteq S_{k}$ be a subgroup. Then
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)=\operatorname{\mathrm{Im}}\left(\widehat{P}^{(n)}_{\mathbf{1}_{G}}\right)$.

Notice that, in the trivial character case, we have the following commutative
diagram of functions, where $\sim_{G}$ is the equivalence relation on
$\operatorname{\mathbb{P}}(V)^{\times k}$ defined by setting
$([u_{1}],\ldots,[u_{k}])\sim_{G}([v_{1}],\ldots,[v_{k}])$ if and only if
$([v_{1}],\ldots,[v_{k}])=([u_{g^{-1}(1)}],\ldots,[u_{g^{-1}(k)}])$, for some
$g\in G$, and $\pi_{G}$ is the canonical projection on the quotient.

${\operatorname{\mathbb{P}}(V)^{\times
k}}$${\operatorname{\mathbb{P}}(V^{\otimes
k})}$${\faktor{\operatorname{\mathbb{P}}(V)^{\times
k}}{\sim_{G}}}$${\operatorname{\mathbb{P}}(V^{\otimes
k}_{\mathbf{1}_{G}})}$$\scriptstyle{\mathrm{seg}^{k,n}}$$\scriptstyle{\pi_{G}}$$\scriptstyle{P^{(n)}_{\mathbf{1}_{G}}}$$\scriptstyle{\mathrm{seg}^{k,n}_{G}}$

The function $\mathrm{seg}^{k,n}_{G}$ is the unique one such that
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}\circ\mathrm{seg}^{k,n}=\mathrm{seg}^{k,n}_{G}\circ\pi_{G}$.
It is an injective function by [24, Theorem 6.60]; moreover
$\operatorname{\mathrm{Im}}(\mathrm{seg}^{k,n}_{G})=\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$.

Our next aim is to prove that the variety
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ has the maximality
property. We define an idempotent function
$p_{\mathbf{1}_{G}}:[n]^{k}\rightarrow B_{\mathbf{1}_{G}}(k,n)$ by setting
$p_{\mathbf{1}_{G}}(x)=\overline{x}$, for all $x\in[n]^{k}$.

######  Proposition 5.3.

The function $p_{\mathbf{1}_{G}}:([n]^{k},\leqslant)\rightarrow
B_{\mathbf{1}_{G}}(k,n)$ is order preserving.

###### Proof.

Let $x,y\in[n]^{k}$ be such that $x\leqslant y$, and $g,h\in G$ such that
$p_{\mathbf{1}_{G}}(x)=g(x)$ and $p_{\mathbf{1}_{G}}(y)=h(y)$. Hence
$p_{\mathbf{1}_{G}}(x)=g(x)\leqslant g(y)=gh^{-1}(p_{\mathbf{1}_{G}}(y))$,
i.e. $p_{\mathbf{1}_{G}}(x)\preccurlyeq p_{\mathbf{1}_{G}}(y)$. ∎

######  Corollary 5.4.

The set $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ has the maximality
property. In particular, $\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}[v]$ is
a $\mathbf{1}_{G}$-matroid, for every
$[v]\in\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$.

###### Proof.

Let $\left[P^{(n)}_{\mathbf{1}_{G}}(v_{1}\otimes....\otimes
v_{k})\right]\in\operatorname{\mathrm{Gr}}_{1_{G}}(k,n)$. By Proposition 4.8,
$\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}[v_{1}\otimes....\otimes v_{k}]$
has a unique maximum $m\in[n]^{k}$. Hence, by Proposition 5.3,
$p_{\mathbf{1}_{G}}(m)$ is the maximum of
$\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}\left[P^{(n)}_{\mathbf{1}_{G}}(v_{1}\otimes....\otimes
v_{k})\right]$. ∎

Let $\rho:[n]^{k}\rightarrow\operatorname{\mathbb{N}}\cup\\{0\\}$ be the rank
function of $[n]^{k}$, i.e. $\rho(x)=\sum_{i=1}^{k}(x_{k}-1)$, for all
$x\in[n]^{k}$. The restriction of $\rho$ provides the rank function of the
poset $B_{\mathbf{1}_{G}}(k,n)$, as the next result shows.

######  Proposition 5.5.

The poset $B_{\mathbf{1}_{G}}(k,n)$ is graded with rank function $\rho$.

###### Proof.

The poset $B_{\mathbf{1}_{G}}(k,n)$ has maximum and minimum, $(n,n,\ldots,n)$
and $(1,1,\ldots,1)$, respectively. Let $x\vartriangleleft y$ in
$B_{\mathbf{1}_{G}}(k,n)$. Then $x\leqslant g(y)$ for some $g\in G$. Let
$z\in[n]^{k}$ be such that $x<z<y$. By Proposition 5.3 we have that
$x\preccurlyeq p_{\mathbf{1}_{G}}(z)\preccurlyeq y$. Since
$\rho(z)=\rho(h(z))$, for all $h\in G$, we have $x\prec
p_{\mathbf{1}_{G}}(z)\prec y$, a contradiction. From this fact we conclude
that $x\vartriangleleft y$ in $B_{\mathbf{1}_{G}}(k,n)$ implies
$\rho(y)=\rho(x)+1$. ∎

######  Remark 5.6.

Notice that Proposition 5.5 follows directly by [18, Lemma 7] since
$B_{\mathbf{1}_{G}}(k,n)$ is a homogeneous quotient of the poset
$B_{\mathbf{1}_{\\{e\\}}}(k,n)$. 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.

######  Example 5.7.

Let $k=n=3$, $A_{3}$ be the alternating subgroup of $S_{3}$. The poset
$B_{\mathbf{1}_{A_{3}}}(3,3)$ has the following Hasse diagram:

$(3,3,3)$$(2,3,3)$$(1,3,3)$$(2,2,3)$$(1,2,3)$$(1,3,2)$$(2,2,2)$$(1,2,2)$$(1,1,3)$$(1,1,2)$$(1,1,1)$

By the weighted Pólya enumeration theorem, one can deduce the rank-generating
function of $B_{\mathbf{1}_{G}}(k,n)$.

######  Proposition 5.8.

The rank-generating function of the poset $B_{\mathbf{1}_{G}}(k,n)$ is

$\sum\limits_{x\in
B_{\mathbf{1}_{G}}(k,n)}q^{\rho(x)}=\frac{1}{|G|}\sum\limits_{g\in
G}\prod\limits_{i=1}^{k}\left([n]_{q^{i}}\right)^{c_{i}(g)},$

where $[n]_{q}:=\sum\limits_{i=0}^{n-1}q^{i}$ is the $q$-analog of $n$ and
$c_{i}(g)$ is the number of cycles of length $i\in[k]$ of the permutation
$g\in S_{k}$.

###### 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 $i\in[n]$. Then the
generating function of the colours is $[n]_{q}$ and
$\rho(x)=\sum_{i=1}^{k}w(x_{i})$, for all $x\in B_{\mathbf{1}_{G}}(k,n)$.
Therefore the generating function of the number of colored arrangements by
weight is the rank-generating function of $B_{\mathbf{1}_{G}}(k,n)$. ∎

######  Corollary 5.9.

Let $G\subseteq S_{k}$ be a group; then the poset $B_{\mathbf{1}_{G}}(k,n)$ is
rank-symmetric.

###### Proof.

Let $B(q):=\frac{1}{|G|}\sum\limits_{g\in
G}\prod\limits_{i=1}^{k}\left([n]_{q^{i}}\right)^{c_{i}(g)}$. Since
$\deg(B(q))=k(n-1)$, the result can be deduced by the easy equality
$q^{k(n-1)}B(q^{-1})=B(q)$. ∎

For the rest of this section, we focus on the case $G=S_{k}$. As for the
Bruhat order on Grassmannian permutations, the poset
$B_{\mathbf{1}_{S_{k}}}(k,n)$ is an induced subposet of $[n]^{k}$ (see [3,
Proposition 4.9] for a similar statement in the Grassmannian context). It is
not difficult to prove that it is a distributive lattice. Then, a direct
consequence of [1, Theorem 3.7 and Proposition 4.2] is that the intervals of
$B_{\mathbf{1}_{S_{k}}}(k,n)$ are shellable.

######  Remark 5.10.

The poset $B_{\mathbf{1}_{G}}(k,n)$ could be not a lattice; see Example 5.7.

In the last result of this section we provide an easy way to produce
$\mathbf{1}_{S_{k}}$-matroids.

######  Proposition 5.11.

Let $[x,y]$ be an interval of $B_{\mathbf{1}_{S_{k}}}(k,n)$; then $[x,y]$ is a
$\mathbf{1}_{S_{k}}$-matroid representable over $\operatorname{\mathbb{C}}$.

###### Proof.

We first prove that $[x,y]$ is representable over $\operatorname{\mathbb{C}}$.
This follows by noting that, if

$[w]:=\left[\left(\sum\limits_{i_{1}=x_{1}}^{y_{1}}e_{i_{1}}\right)\otimes\ldots\otimes\left(\sum\limits_{i_{k}=x_{k}}^{y_{k}}e_{i_{k}}\right)\right]\in\operatorname{\mathrm{Seg}}(k,n),$

then
$\operatorname{\mathrm{supp}}_{\mathbf{1}_{S_{k}}}\left[P^{(n)}_{\mathbf{1}_{S_{k}}}(w)\right]=[x,y]$.
The inclusion
$[x,y]\subseteq\operatorname{\mathrm{supp}}_{\mathbf{1}_{S_{k}}}\left[P^{(n)}_{\mathbf{1}_{S_{k}}}(w)\right]$
is trivial. The other inclusion follows since $x_{1}\leqslant
x_{2}\leqslant\ldots\leqslant x_{k}$ and $y_{1}\leqslant
y_{2}\leqslant\ldots\leqslant y_{k}$, and then the following fact holds: if
$1\leqslant i<j\leqslant k$ and $a>b$ such that $x_{i}\leqslant a\leqslant
y_{i}$, $x_{j}\leqslant b\leqslant y_{j}$, then $x_{i}\leqslant b\leqslant
y_{i}$ and $x_{j}\leqslant a\leqslant y_{j}$. By Corollary 5.4 we have that
$[x,y]$ is a $\mathbf{1}_{S_{k}}$-matroid. ∎

######  Remark 5.12.

Notice that a Bruhat interval $[x,y]$ of Grassmannian permutations is a
matroid (see [3, Section 2.3] and references therein). In general, any Bruhat
interval in a parabolic quotient of a finite Coxeter group is a Coxeter
matroid (see [6]).

The result of Proposition 5.11 is not true for the trivial character of an
arbitrary group. In fact, consider the poset of Example 5.7, and its interval
$[(1,1,3),(1,3,3)]$. Hence, if $\sigma=132\in S_{3}$, we obtain

$\left\\{\sigma^{*}(x):x\in[(1,1,3),(1,3,3)]\right\\}=\left\\{(1,1,2),(1,2,3),(1,3,2),(1,2,2)\right\\}.$

This set has two maxima, $(1,2,3)$ and $(1,3,2)$, so $[(1,1,3),(1,3,3)]$ is
not a $\chi$-matroid; in particular, it is not representable over
$\operatorname{\mathbb{C}}$, because
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{A_{3}}(3,3)}$ has the maximality
property by Corollary 5.4.

##  6 The geometry of $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$

In this section we use ideas from [3] and the combinatorial tools of Section 5
to realize an incidence stratification of the variety
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$. This stratification
provides a set of generators for the Chow $\operatorname{\mathbb{Q}}$-vector
space of $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$, as we prove in
Theorem 6.4. We consider the incidence group over $\operatorname{\mathbb{C}}$
of the poset $B_{\mathbf{1}_{G}}(k,n)$ acting on the projective space
$\operatorname{\mathbb{P}}\left(V^{\otimes k}_{\mathbf{1}_{G}}\right)$. Then,
by [3, Theorem 5.1] the orbits of this action are in bijection with the order
ideals of $B_{\mathbf{1}_{G}}(k,n)$. Given an order ideal
$I\in\mathcal{J}\left(B_{\mathbf{1}_{G}}(k,n)\right)$ let $C_{I}$ be the
corresponding orbit and define

$C^{\mathbf{1}_{G}}_{I}:=C_{I}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n).$

######  Proposition 6.1.

We have that $C^{\mathbf{1}_{G}}_{I}\neq\varnothing$ if and only if $I$ is a
principal order ideal. Hence as posets

$\left\\{I\in\mathcal{J}\left(B_{\mathbf{1}_{G}}(k,n)\right):C^{\mathbf{1}_{G}}_{I}\neq\varnothing\right\\}\simeq
B_{\mathbf{1}_{G}}(k,n).$

###### Proof.

Let $I$ be principal with $m:=\max(I)$. Then
$\left[P^{(n)}_{\mathbf{1}_{G}}(e_{m})\right]\in
C_{I}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$. On the other hand,
let $[v]\in C_{I}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ for
some $v\in V^{\otimes k}_{\mathbf{1}_{G}}$. By Corollary 5.4, we have that
$\max(I)=\max(\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}[v])=\\{m\\}$ for
some $m\in B_{\mathbf{1}_{G}}(k,n)$. The rest of the statement is an immediate
consequence. ∎

For $x\in B_{\mathbf{1}_{G}}(k,n)$, we set
$C_{x}:=C^{\mathbf{1}_{G}}_{x^{\downarrow}}$. By Proposition 6.1, it follows
that

$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)=\biguplus_{x\in
B_{\mathbf{1}_{G}}(k,n)}C_{x}.$

Let $x\in[n]^{k}$ and
$A\in\operatorname{\mathrm{Mat}}_{n,k}(\operatorname{\mathbb{C}})$ be the
matrix of parameters (see Theorem 3.16). We let
$A^{x}\in\operatorname{\mathrm{Mat}}_{n,k}(\operatorname{\mathbb{C}})$ be the
matrix whose entries are $A^{x}_{i,j}=A_{i,j}$ if $i\leqslant x_{j}$, and
$A^{x}_{i,j}=0$ otherwise. We now provide a system of parametric equations for
the algebraic set
$\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$.

######  Theorem 6.2.

Let $x\in B_{\mathbf{1}_{G}}(k,n)$. Then
$\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$
is described by the parametric equations

$\left\\{z_{y}=|G_{y}|^{-1}(\mathbf{1}_{G})_{y,(1,\ldots,k)}(A^{x}):y\in
x^{\downarrow}\right\\}\cup\left\\{z_{y}=0:y\not\in x^{\downarrow}\right\\}.$

###### Proof.

First consider the case $G=\\{e\\}$, the trivial group. Let $x\in[n]^{k}$ and
$[u]\in\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Seg}}(k,n)$;
hence, by Proposition 6.1,
$z:=\max(\operatorname{\mathrm{supp}}_{\mathbf{1}_{G}}[u])\leqslant x$ and
then $v=\sum_{y\leqslant x}(A^{x})^{\otimes k}_{y,(1,\ldots,k)}e_{y}$ is a
representative of $[u]$, for the choice $a_{ij}=0$ whenever $i>z_{i}$, for all
$i\in[n]$ and for all $j\in[k]$.

Let $G\neq\\{e\\}$, $x\in B_{\mathbf{1}_{G}}(k,n)\subseteq[n]^{k}$ and
$[u]\in\overline{C}_{x^{\downarrow
G}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$, $x^{\downarrow
G}\in\mathcal{J}(B_{\mathbf{1}_{G}}(k,n))$. Then, by Proposition 5.3, a
representative of $[u]$ is $w=\frac{1}{|G|}\sum\limits_{g\in G}gv$, for some
$v\in\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Seg}}(k,n)$, where
$x^{\downarrow}\in\mathcal{J}([n]^{k})$ and $\overline{x}=x$.

We claim that, if $x\in[n]^{k}$, then
$p_{\mathbf{1}_{G}}(x^{\downarrow})=\left(p_{\mathbf{1}_{G}}(x)\right)^{\downarrow}$.
First we prove that
$p_{\mathbf{1}_{G}}(x^{\downarrow})\subseteq\left(p_{\mathbf{1}_{G}}(x)\right)^{\downarrow}$.
Let $y\in p_{\mathbf{1}_{G}}(x^{\downarrow})$; then $y=p_{\mathbf{1}_{G}}(z)$
for some $z\leqslant x$. By Proposition 5.3 the function $p_{\mathbf{1}_{G}}$
is order preserving, hence $y=p_{\mathbf{1}_{G}}(z)\preccurlyeq
p_{\mathbf{1}_{G}}(x)$, i.e.
$y\in\left(p_{\mathbf{1}_{G}}(x)\right)^{\downarrow}$. We prove now that
$\left(p_{\mathbf{1}_{G}}(x)\right)^{\downarrow}\subseteq
p_{\mathbf{1}_{G}}(x^{\downarrow})$. Let
$y\in\left(p_{\mathbf{1}_{G}}(x)\right)^{\downarrow}$; then $y\preccurlyeq
p_{\mathbf{1}_{G}}(x)$, i.e. $y\leqslant g(x)$, for some $g\in G$. Therefore
$z:=g^{-1}(y)\leqslant x$. Hence $z\in x^{\downarrow}$ and the result follows
because $y=p_{\mathbf{1}_{G}}(z)$.

Our claim implies that, as sets,

$\mathcal{J}([n]^{k})\ni x^{\downarrow}\simeq\biguplus\limits_{z\preccurlyeq
x}\bigcup\limits_{\\{g\in G:g(z)\leqslant x\\}}\left\\{(z,g(z))\right\\}.$

Therefore, by the previous case and our claim,

$\displaystyle w$ $\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum\limits_{h\in G}h\sum\limits_{y\leqslant
x}(A^{x})^{\otimes k}_{y,(1,\ldots,k)}e_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum\limits_{h\in G}h\sum\limits_{y\preccurlyeq
x}\sum\limits_{\begin{subarray}{c}u\in O_{y}\\\ u\leqslant
x\end{subarray}}(A^{x})^{\otimes k}_{u,(1,\ldots,k)}e_{u}$ $\displaystyle=$
$\displaystyle\frac{1}{|G|}\sum\limits_{y\preccurlyeq
x}\left(\sum\limits_{u\in O_{y}}(A^{x})^{\otimes
k}_{u,(1,\ldots,k)}\right)\sum_{h\in G}e_{h(y)}$ $\displaystyle=$
$\displaystyle\sum\limits_{y\preccurlyeq
x}|G_{y}|^{-1}(\mathbf{1}_{G})_{y,(1,\ldots,k)}(A^{x})P^{(n)}_{\mathbf{1}_{G}}(e_{y}).$

∎

######  Corollary 6.3.

Let $x\in B_{\mathbf{1}_{G}}(k,n)$. Then the projective algebraic set
$\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$
is irreducible. In particular

$\overline{C}_{x}=\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n),$

where $\overline{C}_{x}$ and $\overline{C}_{x^{\downarrow}}$ are the Zariski
closures in $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ and
$\operatorname{\mathbb{P}}\left(V^{\otimes k}_{\mathbf{1}_{G}}\right)$,
respectively.

###### Proof.

As in Proposition 3.2, Theorem 6.2 implies that $\overline{C}_{x}$ is
irreducible. The equality can be deduced by noting that
$C_{x^{\downarrow}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ is an
open set of the projective variety
$\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$.
∎

We have proved in the previous section that the poset
$B_{\mathbf{1}_{G}}(k,n)$ is graded; in the last result of this section we
interpret geometrically its rank function. Moreover we provide a set of
generators for the Chow vector space
$A_{*}\left(\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n);\operatorname{\mathbb{Q}}\right)$
of $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$ over
$\operatorname{\mathbb{Q}}$. Recall from Proposition 5.8 that $c_{i}(g)$ is
the number of cycles of length $i\in[k]$ of the permutation $g\in S_{k}$, and
denote by $\mathrm{HP}_{G}$ the Hilbert-Poincaré polynomial of
$A_{*}\left(\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n);\operatorname{\mathbb{Q}}\right)$.

######  Theorem 6.4.

Let $G\subseteq S_{k}$ be a group and $x\in B_{\mathbf{1}_{G}}(k,n)$. Then:

  1. 1.

$\dim(\overline{C}_{x})=\rho(x)$; in particular
$\dim(\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n))=k(n-1)$.

  2. 2.

$\left\\{[\overline{C}_{x}]:x\in B_{\mathbf{1}_{G}}(k,n)\right\\}$ is a set of
generators of the Chow vector space over $\operatorname{\mathbb{Q}}$ of
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$; in particular

$\mathrm{HP}_{G}(q)\leqslant\frac{1}{|G|}\sum\limits_{g\in
G}\prod\limits_{i=1}^{k}\left([n]_{q^{i}}\right)^{c_{i}(g)},$

where the partial order on polynomials is taken coefficient-wise.

###### Proof.

Let $G=\\{e\\}$. It is easy to verify that
$\overline{C}_{x}\simeq\prod_{i=1}^{k}\operatorname{\mathbb{P}}(\mathbb{C}^{x_{i}})$
as projective varieties. Hence
$\dim(\overline{C}_{x})=\sum\limits_{i\in[k]}(x_{i}-1)=\rho(x)$. In this case
the second statement is clear. In fact
$\operatorname{\mathrm{Seg}}(k,n)\simeq\prod_{i=1}^{k}\operatorname{\mathbb{P}}(\operatorname{\mathbb{C}}^{n})$
as algebraic varieties, and a basis for the Chow vector space of the second is
$\left\\{\prod_{i=1}^{k}[\overline{C}_{x_{i}}]:x\in[n]^{k}\right\\}$; this
basis corresponds to $\left\\{[\overline{C}_{x}]:x\in[n]^{k}\right\\}$ by
using the exterior product isomorphism (see [14, Example 8.3.7]).

Assume now $\\{e\\}\subsetneq G$. The morphism
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}:\operatorname{\mathrm{Seg}}(k,n)\rightarrow\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$
has finite fibers by [24, Theorem 6.60]. Since it is a surjective morphism
between projective varieties, for every subvariety
$X\subseteq\operatorname{\mathrm{Seg}}(k,n)$ of dimension $d$ we have that
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}(X)$ is a $d$-dimensional subvariety of
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$. As we stated in the proof
of Theorem 6.2,
$p_{\mathbf{1}_{G}}(x^{\downarrow})=\left(p_{\mathbf{1}_{G}}(x)\right)^{\downarrow}$;
hence
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}(\overline{C}_{x^{\downarrow}})=\overline{C}_{x^{\downarrow
G}}\subseteq\operatorname{\mathbb{P}}(V^{\otimes k}_{\mathbf{1}_{G}})$. It
follows that

$\widehat{P}^{(n)}_{\mathbf{1}_{G}}\left(\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Seg}}(k,n)\right)\subseteq\widehat{P}^{(n)}_{\mathbf{1}_{G}}(\overline{C}_{x^{\downarrow}})\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)=\overline{C}_{x^{\downarrow
G}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n).$

Therefore, by Corollary 6.3 and the fact that
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}$ preserves the dimensions, we obtain
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}\left(\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Seg}}(k,n)\right)=\overline{C}_{x^{\downarrow
G}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$. This proves the
first statement.

We prove now the second statement. The morphism
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}$ preserves the dimensions and then the
push-forward

$(\widehat{P}^{(n)}_{\mathbf{1}_{G}})_{*}:A_{*}(\operatorname{\mathrm{Seg}}(k,n);\operatorname{\mathbb{Q}})\rightarrow
A_{*}(\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n);\operatorname{\mathbb{Q}})$

is surjective; so the result follows by the $G=\\{e\\}$ case and the fact that
$\widehat{P}^{(n)}_{\mathbf{1}_{G}}(\overline{C}_{x^{\downarrow}}\cap\operatorname{\mathrm{Seg}}(k,n))=\overline{C}_{x^{\downarrow
G}}\cap\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$. The inequality is a
consequence of Proposition 5.8. ∎

For the Segre variety $\operatorname{\mathrm{Seg}}(k,n)$, the strata of the
incidence stratification defined above are smooth. Hence, the incidence
stratification of $\operatorname{\mathrm{Seg}}(k,n)$ we have realized in this
section, is a Seshadri stratification, in the sense of [9, Definition 2.1].
Now we provide an example of incidence stratification which is not Seshadri.

######  Example 6.5.

Let $k=2$, $n=3$ and $G=S_{2}$. By using Theorem 6.2 and Corollary 6.3 for
$x=(3,3)$, and the implicitization method, we obtain that the Chow variety
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{S_{2}}}(2,3)\subseteq\operatorname{\mathbb{P}}\left(V_{\mathbf{1}_{G}}^{\otimes
2}\right)$ is defined by the equation

$x_{13}^{2}x_{22}-x_{12}x_{13}x_{23}+x_{11}x_{23}^{2}+x_{12}^{2}x_{33}-4x_{11}x_{22}x_{33}=0.$

Notice that this is a Brill’s equation, see [4, Section 5.4].

The variety $\overline{C}_{(2,3)}$ has dimension $\rho((2,3))=3$ and
$\overline{C}_{(2,2)}$ is its singular locus, whose defining ideal is
$(x_{13},x_{23},x_{33})+I(\operatorname{\mathrm{Gr}}_{\mathbf{1}_{S_{2}}}(2,3))=(x_{13},x_{23},x_{33})$,
and it has dimension $2$. Hence $\overline{C}_{(2,3)}$ is not smooth in
codimension one, i.e. the incidence stratification $\\{C_{x}:x\in
B_{\mathbf{1}_{S_{2}}}(2,3)\\}$ of
$\operatorname{\mathrm{Gr}}_{\mathbf{1}_{S_{2}}}(2,3)$ is not a Seshadri
stratification.

##  7 Two conjectures and three problems

It is well-known that Schubert varieties of the Grassmannian are Cohen-
Macaulay, see [8], [11] and [19] and reference therein. Supported by several
computations with Macaulay2, we formulate a conjecture about the geometry of
the incidence strata of $\operatorname{\mathrm{Gr}}_{\mathbf{1}_{G}}(k,n)$
defined in the previous section.

######  Conjecture 7.1.

Let $G\subseteq S_{k}$ be a group and $x\in B_{\mathbf{1}_{G}}(k,n)$; then
$\overline{C}_{x}$ is Cohen-Macaulay.

If the techniques based on flat deformations of $LS$ algebras would be
available in our context (see [8]), the previous conjecture could be related
to the following one, which has an independent interest.

######  Conjecture 7.2.

Let $G\subseteq S_{k}$ be a group. Then the order complex of the poset
$B_{\mathbf{1}_{G}}(k,n)$ is shellable.

In Section 4 we defined a poset $B_{\chi}(k,n)$ for one-dimensional characters
$\chi$. In this case, what we know is that if $\chi$ is the alternating
character of $S_{k}$, $B_{\chi}(k,n)$ is the Bruhat order on Grassmannian
permutations, hence it has shellable intervals (see [2, Chapter 2]).

Our conjecture does not extend to other characters, because, for instance, if
$G=\\{e,4321\\}\subseteq S_{4}$ and $\chi(4321)=-1$, then the order complex of
$B_{\chi}(4,3)$ is not shellable. This leads to the following problem.

######  Problem 7.3.

For which one-dimensional characters $\chi$ have we that $B_{\chi}(k,n)$ is
graded? What is the topology of its intervals?

In Proposition 3.3, we provide a condition ensuring that
$\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$ is Zariski closed; this
condition is satisfied in the case of trivial characters (Corollary 5.2). Then
the following problem seems to be natural.

######  Problem 7.4.

For which $f\in\operatorname{\mathrm{End}}\left(V^{\otimes k}\right)$ have we
that $\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$ is Zariski closed, i.e.
$\operatorname{\mathrm{Gr}}_{f}(k,n)=\operatorname{\mathrm{Im}}(\,\widehat{f}\,)$?

Along the paper, we define $\chi$-matroids for a one-dimensional character
$\chi$ of a group $G\subseteq S_{k}$ (see Definition 4.4). We believe that
this notion has an independent interest.

######  Problem 7.5.

Find crypthomorphic definitions of $\chi$-matroids and explore their
combinatorial properties.

##  8 Acknowledgements

We thank Christophe Reutenauer for inspiring the content of Example 4.3. The
first author is grateful to Dipartimento di Matematica of Politecnico di
Milano for its hospitality and financial support.

Both authors are grateful to the amenity of Robecco sul Naviglio for inspiring
some ground ideas of the present work.

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