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In this paper, we provide a recipe for computing Euler number of Grassmann manifold G(k,N) by using Mathai-Quillen formalism (MQ formalism) and Atiyah-Jeffrey construction. Especially, we construct path-integral representation of Euler number of G(k,N). Our model corresponds to a finite dimensional toy-model of topological Yang-Mills theory which motivated Atiyah-Jeffrey construction. As a by-product, we construct free fermion realization of cohomology ring of G(k,N)., 31 pages, Latex, final version accepted for publication in Journal of Geometry and Physics

An S U ( p , q ) -flag domain is an open orbit of the real Lie group S U ( p , q ) acting on the complex flag manifold associated to its complexification S L ( p + q , C ) . Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed S U ( p , q ) -orbit in the ambient manifold. Equivalently, the Borel group contains the AN-factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived.

We define the equivariant Kazhdan–Lusztig polynomial of a matroid equipped with a group of symmetries, generalizing the nonequivariant case. We compute this invariant for arbitrary uniform matroids and for braid matroids of small rank.

We definitively establish that the theory of symmetric Macdonald polynomials aligns with quantum and affine Schubert calculus using a discovery that distinguished weak chains can be identified by chains in the strong (Bruhat) order poset on the type-$A$ affine Weyl group. We construct two one-parameter families of functions that respectively transition positively with Hall-Littlewood and Macdonald's $P$-functions, and specialize to the representatives for Schubert classes of homology and cohomology of the affine Grassmannian. Our approach leads us to conjecture that all elements in a defining set of 3-point genus 0 Gromov-Witten invariants for flag manifolds can be formulated as strong covers., Comment: 29 pages

This paper explores further properties of modules related to Schubert polynomials, introduced by Kraśkiewicz and Pragacz. In this paper we show that any tensor product of Kraśkiewicz–Pragacz modules admits a filtration by Kraśkiewicz–Pragacz modules. This result can be seen as a module-theoretic counterpart of a classical result that the product of Schubert polynomials is a positive sum of Schubert polynomials, and gives a new proof to this classical fact.

Hessenberg varieties are subvarieties of the flag variety parametrized by a linear operator $X$ and a nondecreasing function $h$. The family of Hessenberg varieties for regular $X$ is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and $K$-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on $h$. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on $h$ than here., 14 pages, no figures, v2 contains changes made in response to helpful comments from anonymous referees

We show a Z2-filtered algebraic structure and a “quantum to classical” principle on the torus-equivariant quantum cohomology of a complete flag variety of general Lie type, generalizing earlier works of Leung and the second author. We also provide various applications on equivariant quantum Schubert calculus, including an equivariant quantum Pieri rule for partial flag variety Fln1,⋯,nk;n+1 of Lie type A.

This work studies the flag varieties of p-compact groups, principally through torus-equivariant cohomology, extending methods and tools of classical Schubert calculus and moment graph theory from the setting of real reflection groups to the broader context of complex reflection groups. In particular we give, for the infinite family of p-compact flag varieties corresponding to the complex reflection groups G ( r , 1 , n ) , a generalized GKM characterization (following Goresky–Kottwitz–MacPherson [8] ) of the torus-equivariant cohomology, building an explicit additive basis and showing its relationship with the polynomial or Borel presentation via the localization map.

De Concini, Kac, and Procesi defined a family of subalgebras U q + [ w ] ⊆ U q ( g ) associated with elements w in the Weyl group of a simple Lie algebra g . These algebras are called quantum Schubert cell algebras. We show that, up to a mild cocycle twist, quotients of certain quantum Schubert cell algebras of types E 6 and E 6 ( 1 ) map isomorphically onto distinguished subalgebras of the Faddeev–Reshetikhin–Takhtajan universal bialgebra associated with the braiding on the quantum half-spin representation of U q ( s o 10 ) . We identify the quotients as those obtained by factoring out the quantum Schubert cell algebras by ideals generated by certain submodules with respect to the adjoint action of U q ( s o 10 ) .

We prove that the well-known condition of being a balanced labeling can be characterized in terms of the sliding algorithm on tower diagrams. The characterization involves a generalization of authors@? Rothification algorithm. Using the characterization, we obtain descriptions of Schubert polynomials and Stanley symmetric functions.

Linear sections of Grassmannians provide important examples of varieties. The geometry of these linear sections is closely tied to the spaces of Schubert varieties contained in them. In this paper, we describe the spaces of Schubert varieties contained in hyperplane sections of G(2,n). The group PGL(n) acts with finitely many orbits on the dual of the Plücker space P∗(⋀2V). The orbits are determined by the singular locus of H∩G(2,n). For H in each orbit, we describe the spaces of Schubert varieties contained in H∩G(2,n). We also discuss some generalizations to G(k,n).

This paper develops a new method for studying the cohomology of orthogonal flag varieties. Restriction varieties are subvarieties of orthogonal flag varieties defined by rank conditions with respect to (not necessarily isotropic) flags. They interpolate between Schubert varieties in orthogonal flag varieties and the restrictions of general Schubert varieties in ordinary flag varieties. We give a positive, geometric rule for calculating their cohomology classes, obtaining a branching rule for Schubert calculus for the inclusion of the orthogonal flag varieties in Type A flag varieties. Our rule, in addition to being an essential step in finding a Littlewood–Richardson rule, has applications to computing the moment polytopes of the inclusion of SO ( n ) in SU ( n ) , the asymptotic of the restrictions of representations of SL ( n ) to SO ( n ) and the classes of the moduli spaces of rank two vector bundles with fixed odd determinant on hyperelliptic curves. Furthermore, for odd orthogonal flag varieties, we obtain an algorithm for expressing a Schubert cycle in terms of restrictions of Schubert cycles of Type A flag varieties, thereby giving a geometric (though not positive) algorithm for multiplying any two Schubert cycles.

This paper is the second in a series leading to a type B n geometric Littlewood–Richardson rule. The rule will give an interpretation of the B n Littlewood–Richardson numbers as an intersection of two odd-orthogonal Schubert varieties and will consider a sequence of linear and quadratic deformations of the intersection into a union of odd-orthogonal Schubert varieties. This paper describes the setup for the rule and specifically addresses results for quadratic deformations, including a proof that at each quadratic degeneration, the results occur with multiplicity one. This work is strongly influenced by Vakil’s [14] .

In this paper we prove that a deformed tensor product of two Lefschetz algebras is a Lefschetz algebra. We then use this result in conjunction with some basic Schubert calculus to prove that the coinvariant ring of a finite reflection has the strong Lefschetz property., 31 pages

For positive integers $n>k$, let $P_{n,k}(x)=\displaystyle\sum_{j=0}^k \binom{n}{j}x^j $ be the polynomial obtained by truncating the binomial expansion of $(1+x)^n$ at the $k^{th}$ stage. These polynomials arose in the investigation of Schubert calculus in Grassmannians. In this paper, the authors prove the irreducibility of $P_{n,k}(x)$ over the field of rational numbers when $2\leqslant 2k \leqslant n

We present a partial generalization of the classical Littlewood–Richardson rule (in its version based on Schützenberger's jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin's growth diagrams (for chains in Young's lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S3-symmetric Littlewood–Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.

We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For "reasonable" invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P="singular"; classical pattern avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is insufficient in general. Our approach is analyzed for some common invariants, including Kazhdan-Lusztig polynomials, multiplicity, factoriality, and Gorensteinness, extending [Woo-Yong'05]; the description of the singular locus (which was independently proved by [Billey-Warrington '03], [Cortez '03], [Kassel-Lascoux-Reutenauer'03], [Manivel'01]) is also thus reinterpreted. Our methods are amenable to computer experimentation, based on computing with *Kazhdan-Lusztig ideals* (a class of generalized determinantal ideals) using Macaulay 2. This feature is supplemented by a collection of open problems and conjectures., Comment: 23 pages. Software available at the authors' webpages. Version 2 is the submitted version. It has a nomenclature change: "Bruhat-restricted pattern avoidance" is renamed "interval pattern avoidance"; the introduction has been reorganized

We investigate the line arrangement that results from intersecting d complete flags in C^n. We give a combinatorial description of the matroid T_{n,d} that keeps track of the linear dependence relations among these lines. We prove that the bases of the matroid T_{n,3} characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid T_{n,d}, the triangulations of the product of simplices Delta_{n-1} x \Delta_{d-1}, and the arrangements of d tropical hyperplanes in tropical (n-1)-space. Our work provides a simple and effective criterion to ensure the vanishing of many Schubert structure constants in the flag manifold, and a new perspective on Billey and Vakil's method for computing the non-vanishing ones., Comment: 39 pages, 12 figures, best viewed in color

There are well-known reduction formulas for the universal Schubert coefficients defined on Grassmannians. These coefficients are also known as the Littlewood–Richardson coefficients in the theory of symmetric functions. We restate the reduction formulas combinatorially and provide a combinatorial proof for them.

We study the T-equivariant quantum cohomology of the Grassmannian. We prove the vanishing of a certain class of equivariant quantum Littlewood-Richardson coefficients, which implies an equivariant quantum Pieri rule. As in the equivariant case, this implies an algorithm to compute the equivariant quantum Littlewood-Richardson coefficients., 24 pages, LaTeX

The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds G/B in terms of patterns in root systems. We generalize Lakshmibai–Sandhya's well-known result that says that a Schubert variety in SL(n)/B is smooth if and only if the corresponding permutation avoids the patterns 3412 and 4231. Our criterion is formulated uniformly in general Lie theoretic terms. We define a notion of pattern in Weyl group elements and show that a Schubert variety is smooth (or rationally smooth) if and only if the corresponding element of the Weyl group avoids a certain finite list of patterns. These forbidden patterns live only in root subsystems with star-shaped Dynkin diagrams. In the simply-laced case the list of forbidden patterns is especially simple: besides two patterns of type A3 that appear in Lakshmibai–Sandhya's criterion we only need one additional forbidden pattern of type D4. In terms of these patterns, the only difference between smoothness and rational smoothness is a single pattern in type B2. Remarkably, several other important classes of elements in Weyl groups can also be described in terms of forbidden patterns. For example, the fully commutative elements in Weyl groups have such a characterization. In order to prove our criterion we used several known results for the classical types. For the exceptional types, our proof is based on computer verifications. In order to conduct such a verification for the computationally challenging type E8, we derived several general results on Poincaré polynomials of cohomology rings of Schubert varieties based on parabolic decomposition, which have an independent interest.

We propose a new approach to the multiplication of Schubert classes in the K-theory of the flag variety. This extends the work of Fomin and Kirillov in the cohomology case, and is based on the quadratic algebra defined by them. More precisely, we define K-theoretic versions of the Dunkl elements considered by Fomin and Kirillov, show that they commute, and use them to describe the structure constants of the K-theory of the flag variety with respect to its basis of Schubert classes.

Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After discussing the connection between Horn's Problem and Nielsen's Theorem, we move on to characterizing the eigenvalues of the partial trace of a matrix., 40 pages. Accepted for publication in Annals of Physics

We obtain a tableau definition of the skew Schubert polynomials named by Lascoux, which are defined as flagged double skew Schur functions. These polynomials are in fact Schubert polynomials in two sets of variables indexed by 321-avoiding permutations. From the divided difference definition of the skew Schubert polynomials, we construct a lattice path interpretation based on the Chen–Li–Louck pairing lemma. The lattice path explanation immediately leads to the determinantal definition and the tableau definition of the skew Schubert polynomials. For the case of a single variable set, the skew Schubert polynomials reduce to flagged skew Schur functions as studied by Wachs and by Billey, Jockusch, and Stanley. We also present a lattice path interpretation for the isobaric divided difference operators, and derive an expression of the flagged Schur function in terms of isobaric operators acting on a monomial. Moreover, we find lattice path interpretations for the Giambelli identity and the Lascoux–Pragacz identity for super-Schur functions. For the super-Lascoux–Pragacz identity, the lattice path construction is related to the code of the partition which determines the directions of the lines parallel to the y-axis in the lattice.

We construct a family of functors assigning an R-module to a flag of R-modules, where R is a commutative ring. As particular instances, we get flagged Schur functors and Schubert functors, the latter family being indexed by permutations. We identify Schubert functors for vexillary permutations with some flagged Schur functors, thus establishing a functorial analogue of a theorem of Lascoux and Schützenberger from C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 447 and of Wachs from J. Combin. Theory Ser. A 40 (1985) 276. Over an infinite field, we study the trace of a Schubert module, which is a cyclic module over a Borel subgroup B, restricted to the maximal torus. The main result of the paper says that this trace is equal to the corresponding Schubert polynomial of Lascoux and Schützenberger (C. R. Acad. Sci. Paris Sér. I Math. 294 (1982) 447). We also investigate filtrations of B-modules associated with the Monk formula (Proc. London Math. Soc. 9 (1959) 253) and transition formula from Lett. Math. Phys. 10 (1985) 111.

Let $G$ denote an adjoint semi-simple group over an algebraically closed field and $T$ a maximal torus of $G$. Following Contou-Carr\`ere [CC], we consider the Bott-Samelson resolution of a Schubert variety as a variety of galleries in the building associated to the group $G$. We first determine a cellular decomposition of this variety analogous to the Bruhat decomposition of a Schubert variety and then we describe the fibre of this resolution above a $T-$fixed point., Comment: 17 pages

Answering a question raised by S. Friedland, we show that the possible eigenvalues of Hermitian matrices (or compact operators) A, B, and C with C⩽A+B are given by the same inequalities as in Klyachko's theorem for the case where C=A+B , except that the equality corresponding to tr (C)= tr (A)+ tr (B) is replaced by the inequality corresponding to tr (C)⩽ tr (A)+ tr (B) . The possible types of finitely generated torsion modules A, B, and C over a discrete valuation ring such that there is an exact sequence B→C→A are characterized by the same inequalities.

We study algebraic aspects of equivariant quantum cohomology algebra of the flag manifold. We introduce and study the quantum double Schubert polynomials S w (x,y) , which are the Lascoux–Schutzenberger type representatives of the equivariant quantum cohomology classes. Our approach is based on the quantum Cauchy identity. We define also quantum Schubert polynomials S w (x) as the Gram–Schmidt orthogonalization of some set of monomials with respect to the scalar product, defined by the Grothendieck residue. Using quantum Cauchy identity, we prove that S w (x)= S w (x,y)| y=0 and as a corollary obtain a simple formula for the quantum Schubert polynomials S w (x)=∂ ww 0 (y) S w 0 (x,y)| y=0 . We also prove the higher genus analog of Vafa–Intriligator's formula for the flag manifolds and study the quantum residues generating function. We introduce the Ehresmann–Bruhat graph on the symmetric group and formulate the equivariant quantum Pieri rule.

We prove the results on the tangent spaces to Schubert varieties announced in [V. Lakshmibai, Math. Res. Lett.2 (1995), 473–477] for G classical. We give two descriptions of the tangent space to a Schubert variety at id. The first description is in terms of the root system, and the second one is in terms of multiplicities of certain weights in the fundamental representations of G.

We characterize the harmonic forms on a flag manifold K / T defined by Kostant in 1963 in terms of a Poisson structure. Namely, they are “Poisson harmonic” with respect to the so-called Bruhat Poisson structure on K / T . This enables us to give Poisson geometrical proofs of many of the special properties of these harmonic forms. In particular, we construct explicit representatives for the Schubert basis of the S 1 -equivariant cohomology of K / T , where the S 1 -action is defined by ρ . Using a simple argument in equivariant cohomology, we recover the connection between the Kostant harmonic forms and the Schubert calculus on K / T that was found by Kostant and Kumar in 1986. By using a family of symplectic structures on K / T , we also show that the Kostant harmonic forms are limits of the more familiar Hodge harmonic forms with respect to a family of Hermitian metrics on K / T .

In analogy with transition equations for type A Schubert polynomials given by Lascoux and Schutzenberger (1982), we give recursive formulas for computing representatives of the Schubert classes for the isotropic flag manifolds. These representatives are exactly the Schubert polynomials found in Billey and Haiman (1995). This new approach to finding Schubert polynomials is very closely related to the geometry of the flag manifold and has the advantage that it does not require explicit computations with divided difference operators. The generalized transition equations also lead to a recursion for Stanley symmetric functions and a new proof of Chevalley's intersection formula for Schubert varieties. The proofs involve a careful study of the Bruhat order for the Weyl groups and two simple lemmas for applying divided difference operators.

Letwbe an element of the Weyl groupSn, and letXwbe the Schubert variety associated towin the flag manifoldSLn( C )/B. Lakshmibai and Sandhya showed thatXwis smooth if and only ifwavoids the patterns 4231 and 3412. Using two tests for rational smoothness from Carrell and Peterson, we show that rational smoothness ofXwis characterized by pattern avoidance for typesBandCas well. A key step in the proof of this result is a sequence of rules for factoring the Poincare polynomials for the cohomology ring ofXw, generalizing the work of Gasharov. Patterns can also be used to characterize actual smoothness of Schubert varieties

Motivated by the recent discovery of a simple quantization procedure for Schubert polynomials we study the expansion of Schur and Schubert polynomials into standard elementary monomials (SEM). The SEM expansion of Schur polynomials can be described algebraically by a simple variant of the Jacobi–Trudi formula and combinatorially by a rule based on posets of staircase box diagrams. These posets are seen to be rank symmetric and order isomorphic to certain principal order ideals in the Bruhat order of symmetric groups ranging between the full symmetric group and the respective maximal Boolean sublattice. We prove and conjecture extensions of these results for general Schubert polynomials. The featured conjectures are: (1) an interpretation of SEM expansions as “alternating approximations” and (2) surprising properties of different numbers naturally associated to SEM expansions. This hints at as yet undiscovered deeper symmetry properties of the SEM expansion of Schubert polynomials.

Finding a combinatorial rule for the multiplication of Schubert polynomials is a long standing problem. In this paper we give a combinatorial proof of the extended Pieri rule as conjectured by N. Bergeron and S. Billey, which says how to multiply a Schubert polynomial by a complete or elementary symmetric polynomial, and describe some observations in the direction of a general rule.

We give an elementary proof of the following known fact: any multihomogenous component in the homogeneous coordinate ring of a Schubert variety insideGLn/Bhas basis given by the standard monomials 6 .

We tell the story of Schubert Polynomials.RésuméNous racontons l'histoire des polynômes de Schubert.

In these notes, we present some counter-examples of two propositions concerning cellular decompositions occuring in a Bott-Samelson variety [G, Proposition 4 and Proposition 7]. Then, we prove that the description of the fibre of a Bott-Samelson resolution over a point of a Schubert variety is still valid in the Kac-Moody setting.

Schubert polynomials were introduced and extensively developed by Lascoux and Schutzenberger, after an earlier less combinatorial version had been considered by Bernstein, Gelfand and Gelfand and Demazure. We give a new development of the theory of Schubert polynomials based on formal computations in the algebra of operators u1, u2, ... satisfying the relations u2i=0, uiuj=ujui if |i−j| ≥ 2, and uiui+1ui = ui + 1uiui + 1. We call this algebra the nilCoxeter algebra of the symmetric group S n. Our development leads to simple proofs of many standard results, in particular, (a) symmetry of the "stable Schubert polynomials" Fw, (b) an explicit combinatorial formula for Schubert polynomials due to Billey, Jockusch and Stanley, (c) the " Cauchy formula" for Schubert polynomials, and (d) a formula of Macdonald for S w,(1, 1, ...). Our main new result is a proof of a conjectured q-analogue of (d), due to Macdonald which gives a formula for S w(1, q, q2,...).

Two points are discussed in this note. First, it is shown that in the case of finite Weyl groups, a proof of a conjecture of Deodhar concerning Bruhat intervals can be derived from elementary properties of Schubert varieties. Second, a description of the Zariski tangent spaces of these varieties in some representation-theoretic terms is given, and some consequences of this description are then derived.

We show a combinatorial rule based on diagrams (finite nonempty sets of lattice points (i, j) in the positive quadrant) for the construction of the Schubert polynomials. In the particular case where the Schubert polynomial is a Schur function we give a bijection between our diagrams and column strict tableaux. A different algorithm had been conjectured (and proved in the case of vexillary permutations) by A. Kohnert (Ph.D. dissertation, Universitat auf Bayreuth, 1990). We give, at the end of this paper, a sketch of how one would show the equivalence of the two rules.

We compute the filling radius with rational coefficients of C P n by studying the Serre spectral sequence of the total space of the unit tangent bundle viewed as a principal SO(3)-bundle on the Grassmannian of 2-planes in C P n . We also compute the (integer) filling radius of C P n and exhibit a torsion obstruction to filling C P n . We discuss a related aspect of the global geometry of C P n .

Schubert subvarieties of Grassmannians and, more generally, varieties G/P (with G a linear algebraic group and P a parabolic subgroup) have long been a rich source for algebraic geometers and topologists. Recent interest has focussed on their singularities, which are of interest due to the relationship between the intersection homological properties of a singularity of a Schubert variety in G/P and invariants of composition series of Verma modules [KL, GM]. This work presents a simple combinatorial algorithm for deciding whether a Schubert variety in G/P, where G = SL,, is singular. This leads to a geometric characterization of the nonsingular Schubert varieties, as sequences of Grassmannian bundles over Grassmannians. The combinatorial techniques are those of [P] (of whose existence the author was unaware). A more general result of this nature has been proved by Seshadri and Lakshmibai [LakS], who use Standard Monomial Theory to find the ideal of the singular locus of a Schubert variety in G/B, where G is an algebraic group with a Standard Monomial Theory, and B is a Bore1 subgroup of G. Ryan [R] has obtained results similar to ours, using general techniques for understanding determinantal varieties. The present work has the advantage of being quite elementary, while sharing aspects of both of the above. Or special interest here are the combinatorial techniques, which appear to be a basic connection between the singularities of Schubert Varieties, the geometry of the tibres of Springer’s resolution of the singularities of the nilpotent scheme (see [Wl ] ), and the Kazhdan-Lusztig polynomials. A related note [W2] discusses some applications of these techniques to the computation of the Kazhdan-Lusztig polynomials.

The precise determination of the nature and form of system invariants under output feedback is of fundamental interest in classical and modern control theory. An algebraic solution of this problem would entail, for example, a description of the generators and relations of the ring of output feedback invariants. Geometrically, we seek to describe the basic properties of the quotient space whose points consist of feedback equivalence classes of the system, recovering the algebraic theory in terms of the function theory on this space. Naturally, any description should be made in system theoretic terms. In this paper we construct and study, in the scalar case, the quotient manifold for linear systems modulo the full output feedback group using classical algebraic geometry. As a corollary, we obtain explicit generators for the associated ring of invariants which we then interpret in terms of classical control theory, e.g. in terms of root-locus plots. Turning to the question of the existence of continuous canonical forms, we show that these exist globally, only when the system order is odd. In the even case, the obstruction to the existence of globally defined continuous forms is described in terms of the values of the system Cauchy index. This is illustrated for low order examples and certain questions concerning relations among the generators is treated using methods of the Schubert calculus. The complete set of invariants for the full output feedback group is obtained and our results concerning the existence and nonexistence of canonical forms is also compared to related results contained in the literature.