Del Fra Research Articles

Neorealismo e antropologia: Visconti e De Martino in Lucania

In the late nineteen fifties in Lucania, an area that became symbolic as regards the peasant condition which would be redeemed and changed through new programmes and politics, Ernesto De Martino studies and records the ancient mythical and ritualistic performance of the game of the sickle. In the same period Luchino Visconti studies the location and makes a detailed research for Rocco and his brothers, one of the masterpieces of Italian neo-realism, released in Italy in 1960. That same year L. Del Fra presents A passion for wheat, a documentary based on the documents and archive material of the research conducted by De Martino. Visconti's neo-realism and the studies in southern Italy are connected by the same methodology used and the creation of the paradigms of the first anthropological cinema or documentary which in Italy is, in its beginnings, the meridional cinema.

Partition identities and quiver representations

We present a particular connection between classical partition combinatorics and the theory of quiver representations. Specifically, we give a bijective proof of an analogue of A. L. Cauchy’s Durfee square identity to multipartitions. We then use this result to give a new proof of M. Reineke’s identity in the case of quivers \({\mathcal {Q}}\) of Dynkin type A. Our identity is stated in terms of the lacing diagrams of S. Abeasis–A. Del Fra, which parameterize orbits of the representation space of \({\mathcal {Q}}\) for a fixed dimension vector.

Bilinear dual hyperovals in the largest ambient spaces

The following facts are shown for a bilinear dual hyperoval $${\mathcal {S}}$$ of rank n. The ambient space $${\mathbf {A}}({\mathcal {S}})$$ has vector dimension at most $$n(n+1)/2$$ . The dimension of $${\mathbf {A}}({\mathcal {S}})$$ is $$n(n+1)/2$$ if and only if $${\mathcal {S}}$$ is isomorphic to the Huybrechts or the Buratti–Del Fra dual hyperoval.

On some bilinear dual hyperovals

It is shown in Yoshiara (2004) that, if d-dimensional dual hyperovals exist in V(n,2) (GF(2)-vector space of rank n), then 2d+1≤n≤(d+1)(d+2)/2+2, and conjectured that n≤(d+1)(d+2)/2. Known bilinear dual hyperovals in V((d+1)(d+2)/2,2) are the Huybrechts dual hyperoval and the Buratti–Del Fra dual hyperoval. In this paper, we investigate on the covering map π:Sc′(l′,GF(2r′))→Sc(l,GF(2r)), where the dual hyperovals Sc′(l′,GF(2r′)) and Sc(l,GF(2r)) are constructed in Taniguchi (2014). Using the result, we show that the Buratti–Del Fra dual hyperoval has a bilinear quotient in V(2d+1,2) if d is odd. On the other hand, we show that the Huybrechts dual hyperoval has no bilinear quotient in V(2d+1,2). We also determine the automorphism group of Sc(l,GF(2r)), and show that Aut(Sc(l2,GF(2rl1)))<Aut(Sc(l,GF(2r))).

New dimensional dual hyperovals, which are not quotients of the classical dual hyperovals

Let d≥2. If d-dimensional dual hyperoval exists in V(n,2) (n-dimensional vector space over GF(2)), then 2d+1≤n≤(d+1)(d+2)/2+2 (Yoshiara [15], 2004), and it is conjectured that n≤(d+1)(d+2)/2. In V((d+1)(d+2)/2,2), there are four known non-isomorphic d-dimensional dual hyperovals. These are the Huybrechts dual hyperoval (Huybrechts [5], 2002), the Buratti–Del Fra dual hyperoval (Buratti–Del Fra [1], 2003), (Del Fra and Yoshiara [2], 2005), (Taniguchi and Yoshiara [13], 2012), the Veronesean dual hyperoval (Thas and van Maldeghem [14], 2004, Yoshiara [15], 2004) and the deformation of Veronesean dual hyperoval (Taniguchi [9], 2009). Many of the known dual hyperovals in V(n,2) for 2d+2<n<(d+1)(d+2)/2 are quotients of the four examples in V((d+1)(d+2)/2,2). Very recently, we constructed examples of d-dimensional dual hyperovals for n=3d+3 with d≥3, n=4d−2 with d≥4, and n=3d+1 with 3≤d≤13 satisfying some conditions (Taniguchi [11], 2013), which are not quotients of the four examples in V((d+1)(d+2)/2,2). In this paper, we modify the method of constructing the Buratti–Del Fra dual hyperoval (Taniguchi and Yoshiara [13], 2012) combined with Example 3.4 of Dempwolff [3] (2013), and construct many non-isomorphic symmetric bilinear dual hyperovals in V(((1/r)d2+3d+2)/2,2) for d=lr≥4, which are not quotients of the four dual hyperovals in V((d+1)(d+2)/2,2).

A unified description of four simply connected dimensional dual hyperovals

A unified and concise description of the following simply connected dimensional dual hyperovals is obtained: the Veronesean, the Huybrechts, the Buratti–Del Fra and the Taniguchi dual hyperovals of rank n over F2.

Simple expressions of the Buratti–Del Fra dual hyperoval and the deformation of the Veronesean dual hyperoval

Abstract The concept of a dimensional dual hyperoval was introduced by Huybrechts and Pasini in 1999 in [C. Huybrechts and A. Pasini, Frag-transitive extensions of dual affine spaces, Beitrage Algebra Geom. 40 (1999), 503–532]. It is conjectured in [S. Yoshiara, Ambient spaces of dimensional dual arcs, Journal of Algebraic Combinatorics 19 (2004), 5–23] that, if a d-dimensional dual hyperoval exists in P G ( n , 2 ) , then 2 d ⩽ n ⩽ d ( d + 3 ) / 2 . Known d-dimensional dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) are (a) Huybrechtsʼ dual hyperoval [C. Huybrechts, Dimensional dual hyperovals in projective spaces and c.AC* geometries, Discrete Math. 255 (2002), 503–532], (b) Buratti and Del Fraʼs dual hyperoval [M. Buratti and A. Del Fra, Semi-Boolean quadruple systems and dimensional dual hyperovals, Adv. Geom. 3 (2003), 245–253], [A. Del Fra and S. Yoshiara, Dimensional dual hyperovals associated with Steiner systems, European J. Combin. 26 (2005), 173–194], (c) the Veronesean dual hyperoval [J. Thas and H. van Maldeghem, Characterizations of the finite quadric Veroneseans V n 2 n , The Quarterly Journal of Mathematics, Oxford, 55 (2004), 99–113], [S. Yoshiara, Ambient spaces of dimensional dual arcs, Journal of Algebraic Combinatorics 19 (2004), 5–23], and (d) the deformation of the Veronesean dual hyperoval [H. Taniguchi, A new family of dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) with d ⩾ 3 , Discrete Mathematics 309 (2009), 418–429]. (b) and (d) are originally constructed using complicated expressions, so it has been difficult to deal with these dual hyperovals until now. In this article, we present simple expressions of (b) and (d), which enable us to calculate these dual hyperovals easily.

A new construction of the [formula omitted]-dimensional Buratti–Del Fra dual hyperoval

The Buratti–Del Fra dual hyperoval Dd(F2) is one of the four known infinite families of simply connected d-dimensional dual hyperovals over F2 with ambient space of vector dimension (d+1)(d+2)/2 (Buratti and Del Fra (2003) [1]). A criterion (Proposition 1) is given for a d-dimensional dual hyperoval over F2 to be covered by Dd(F2) in terms of the addition formula. Using it, we provide a simpler model of Dd(F2) (Proposition 3). We also give conditions (Lemma 4) for a collection S[B] of (d+1)-dimensional subspaces of K⊕K constructed from a symmetric bilinear form B on K≅F2d+1 to be a quotient of Dd(F2). For when d is even, an explicit form B satisfying these conditions is given. We also provide a proof for the fact that the affine expansion of Dd(F2) is covered by the halved hypercube (Proposition 10).

Quotients of the deformation of Veronesean dual hyperoval in [formula omitted

Let d ≥ 3 . In P G ( d ( d + 3 ) / 2 , 2 ) , there are four known non-isomorphic d -dimensional dual hyperovals by now. These are Huybrechts’ dual hyperoval by Huybrechts (2002) [4], Buratti-Del Fra’s dual hyperoval by Buratti and Del Fra (2003) [1], Del Fra and Yoshiara (2005) [3], Veronesean dual hyperoval by Thas and van Maldeghem (2004) [9], Yoshiara (2004) [12] and the dual hyperoval, which is a deformation of Veronesean dual hyperoval by Taniguchi (2009) [6]. In this paper, using a generator σ of the Galois group G a l ( G F ( 2 d m ) / G F ( 2 ) ) for some m ≥ 3 , we construct a d -dimensional dual hyperoval T σ in P G ( 3 d , 2 ) , which is a quotient of the dual hyperoval of [6]. Moreover, for generators σ , τ ∈ G a l ( G F ( 2 d m ) / G F ( 2 ) ) , if T σ and T τ are isomorphic, then we show that σ = τ or σ = τ − 1 on G F ( 2 d ) . Hence, we see that there are many non-isomorphic quotients in P G ( 3 d , 2 ) for the dual hyperoval of [6] if d is large.

On [formula omitted]-dimensional Buratti–Del Fra type dual hyperovals in [formula omitted

Let d ≥ 3 . In P G ( d ( d + 3 ) / 2 , 2 ) , there are four known non-isomorphic d -dimensional dual hyperovals by now. These are Huybrechts’ dual hyperoval (Huybrechts (2002) [6]), Buratti–Del Fra’s dual hyperoval (Buratti and Del Fra (2003) [1], Del Fra and Yoshiara (2005) [3]), Veronesean dual hyperoval (Thas and van Maldeghem (2004) [9], Yoshiara (2004) [12]), and the dual hyperoval which is a deformation of Veronesean dual hyperoval (Taniguchi (2009) [8]). Using quadratic APN functions on G F ( 2 d + 1 ) , Yoshiara (2009) [10], constructed d -dimensional dual hyperovals in P G ( 2 d + 1 , 2 ) . This construction enables us to investigate quadratic APN functions from the view point of dual hyperovals (Edel (2009) [4]). Yoshiara generalized this construction in Yoshiara (2008) [11]. Note that these Yoshiara’s dual hyperovals are quotients of Huybrechts’ dual hyperoval. In this note, using quadratic APN functions on G F ( 2 d ) , we construct d -dimensional dual hyperovals in P G ( 3 d , 2 ) , which are quotients of Buratti–Del Fra’s dual hyperoval. Moreover, we prove that, if two of these dual hyperovals are isomorphic, then the corresponding quadratic APN functions are extended affine equivalent. We call these dual hyperovals Buratti–Del Fra type dual hyperovals. Then, if d is sufficiently large, there are many non-isomorphic d -dimensional Buratti–Del Fra type dual hyperovals in P G ( 3 d , 2 ) .

On some d-dimensional dual hyperovals in [formula omitted

Let d ≥ 3 . Let H be a d + 1 -dimensional vector space over G F ( 2 ) and { e 0 , … , e d } be a specified basis of H . We define S u p p ( t ) ≔ { e t 1 , … , e t l } , a subset of a specified base for a non-zero vector t = e t 1 + ⋯ + e t l of H , and S u p p ( 0 ) ≔ 0̸ . We also define J ( t ) ≔ S u p p ( t ) if | S u p p ( t ) | is odd, and J ( t ) ≔ S u p p ( t ) ∪ { 0 } if | S u p p ( t ) | is even. For s , t ∈ H , let { a ( s , t ) } be elements of H ⊕ ( H ∧ H ) which satisfy the following conditions: (1) a ( s , s ) = ( 0 , 0 ) , (2) a ( s , t ) = a ( t , s ) , (3) a ( s , t ) ≠ ( 0 , 0 ) if s ≠ t , (4) a ( s , t ) = a ( s ′ , t ′ ) if and only if { s , t } = { s ′ , t ′ } , (5) { a ( s , t ) | t ∈ H } is a vector space over G F ( 2 ) , (6) { a ( s , t ) | s , t ∈ H } generate H ⊕ ( H ∧ H ) . Then, it is known that S ≔ { X ( s ) | s ∈ H } , where X ( s ) ≔ { a ( s , t ) | t ∈ H ∖ { s } } , is a dual hyperoval in P G ( d ( d + 3 ) / 2 , 2 ) = ( H ⊕ ( H ∧ H ) ) ∖ { ( 0 , 0 ) } . In this note, we assume that, for s , t ∈ H , there exists some x s , t in G F ( 2 ) such that a ( s , t ) satisfies the following equation: a ( s , t ) = ∑ w ∈ J ( t ) a ( s , w ) + x s , t ( a ( s , 0 ) + a ( s , e 0 ) ) . Then, we prove that the dual hyperoval constructed by { a ( s , t ) } is isomorphic to either the Huybrechts’ dual hyperoval, or the Buratti and Del Fra’s dual hyperoval.

A new family of dual hyperovals in [formula omitted] with [formula omitted

In 2003, a class of d -dimensional dual hyperovals was constructed by Buratti and Del Fra [M. Buratti, A. Del Fra, Semi-Boolean quadruple systems and dimensional dual hyperovals, Adv. Geom. 3 (2003) 245–253] based on Huybrechts’ dual hyperovals. We construct a family based on Veronesean dual hyperovals in PG ( d ( d + 3 ) / 2 , 2 ) . However, these dual hyperovals are not isomorphic to the Veronesean dual hyperovals and do not satisfy Property ( T ) . Hence, we conclude that these are new dual hyperovals.

A Note on Cyclic m-cycle Systems of Kr(m)

It was proved by Buratti and Del Fra that for each pair of odd integers r and m, there exists a cyclic m-cycle system of the balanced complete r-partite graph Kr(m) except for the case when r=m=3. In this note, we study the existence of a cyclic m-cycle system of Kr(m) where r or m is even. Combining the work of Buratti and Del Fra, we prove that cyclic m-cycle systems of Kr(m) exist if and only if (a) Kr(m) is an even graph (b) (r, m)≠ (3, 3) and (c) (r,m)≢ (t , 2) (mod 4) where t ∈ {2,3}.

On d-dual hyperovals in

Abstract In P G ( d ( d + 3 ) / 2 , 2 ) , there are three known families of dual hyperovals; Huybrechts's dual hyperovals ([C. Huybrechts, Dimensional dual hyperovals in projective spaces and c.AC* geometries, Discrete math. 255 (2002), 503–532]), Veronesean dual hyperovals ([J. Thas and H. van Maldeghem, Characterizations of the finite quadric Veroneseans V n 2 n , Quart. J. Math. Oxford. 55 (2004), 99–113] and [S. Yoshiara, Ambient spaces of dimensional dual arcs, J. Alg. Combin. 19 (2004), 5–23]), and the family constructed by Buratti and Del Fra ([M. Buratti and A. del Fra, Semi-Boolean quadruple systems and dimensional dual hyperovals, Adv. Geom. 3 (2003), 245–253] and [A. del Fra and S. Yoshiara, Dimensional dual hyperovals associated with Steiner systems, Europ. J. Combinatorics. 26 (2005), 173–194]) based on Huybrechts's dual hyper-ovals. In this note, we construct a family of dual hyperovals based on Veronesean dual hyperovals. We also prove that our dual hyperovals are not isomorphic to the Veronesean dual hyperovals and do not satisfy Property (T ). Hence, we have a new family of dual hyperovals in P G ( d ( d + 3 ) / 2 , 2 ) .

Four positive formulae for type A quiver polynomials

We give four positive formulae for the (equioriented type A) quiver polynomials of Buch and Fulton [BF99 ]. All four formulae are combinatorial, in the sense that they are expressed in terms of combinatorial objects of certain types: Zelevinsky permutations, lacing diagrams, Young tableaux, and pipe dreams (also known as rc-graphs). Three of our formulae are multiplicity-free and geometric, meaning that their summands have coefficient 1 and correspond bijectively to components of a torus-invariant scheme. The remaining (presently non-geometric) formula is a variant of the conjecture of Buch and Fulton in terms of factor sequences of Young tableaux [BF99 ]; our proof of it proceeds by way of a new characterization of the tableaux counted by quiver constants. All four formulae come naturally in “doubled” versions, two for double quiver polynomials, and the other two for their stable limits, the double quiver functions, where setting half the variables equal to the other half specializes to the ordinary case. Our method begins by identifying quiver polynomials as multidegrees [BB82 , Jos84 , BB85 , Ros89 ] via equivariant Chow groups [EG98 ]. Then we make use of Zelevinsky’s map from quiver loci to open subvarieties of Schubert varieties in partial flag manifolds [Zel85 ]. Interpreted in equivariant cohomology, this lets us write double quiver polynomials as ratios of double Schubert polynomials [LS82 ] associated to Zelevinsky permutations; this is our first formula. In the process, we provide a simple argument that Zelevinsky maps are scheme-theoretic isomorphisms (originally proved in [LM98 ]). Writing double Schubert polynomials in terms of pipe dreams [FK96 ] then provides another geometric formula for double quiver polynomials, via [KM05 ]. The combinatorics of pipe dreams for Zelevinsky permutations implies an expression for limits of double quiver polynomials in terms of products of Stanley symmetric functions [Sta84 ]. A degeneration of quiver loci (orbit closures of GL on quiver representations) to unions of products of matrix Schubert varieties [Ful92 , KM05 ] identifies the summands in our Stanley function formula combinatorially, as lacing diagrams that we construct based on the strands of Abeasis and Del Fra in the representation theory of quivers [AD80 ]. Finally, we apply the combinatorial theory of key polynomials to pass from our lacing diagram formula to a double Schur function formula in terms of peelable tableaux [RS95a , RS98 ], and from there to our formula of Buch–Fulton type.

A geometric characterization of the groups Suz and HS

Buekenhout and Hubaut [BH] studied finite connected c.C,-geometries (more generally, c.C,,-geometries) r whose point-residues are classical quadrangles, admitting a flag-transitive group G whose point stabilizer induces an automorphism group containing the corresponding classical simple group. In their Lemma 7.3, however, they overlooked the group M,, as a transitive extension of M,, acting on the 10 points of a line in a U,(3)quadrangle H,(3’). There is, in fact, an interesting c.C,-geometry associated with the sporadic Suzuki group Suz with such point residues (see [BF], [Bu]). This is a 3-local geometry; it was described in [St]; [Ro, 7.2, pp,326-3271. The set 9 is a certain conjugacy class of subgroups of order 3 in SUZ, and the sets 9 and 9 consist of pairs and maximal sets of mutually commuting subgroups in 9”, respectively. Recently, A. Del Fra, D. Ghinelli, T. Meixner, and A. Pasini [FGMP] studied c.C,-geometries whose point-residues are classical thick quadrangles, admitting a flag-transitive group G. They succeeded in classifying these geometries, except in the case where point-residues are again isomorphic to the U,(3)-quadrangle H,(3’).