A geometric approach to alternating <Emphasis Type="Italic">k</Emphasis>-linear forms

Denote by \({{\mathcal {G}}}_k(V)\) the Grassmannian of the k-subspaces of a vector space V over a field \({\mathbb {K}}\). There is a natural correspondence between hyperplanes H of \({\mathcal {G}}_k(V)\) and alternating k-linear forms on V defined up to a scalar multiple. Given a hyperplane H of \({{\mathcal {G}}_k}(V)\), we define a subspace \(R^{\uparrow }(H)\) of \({{\mathcal {G}}_{k-1}}(V)\) whose elements are the \((k-1)\)-subspaces A such that all k-spaces containing A belong to H. When \(n-k\) is even, \(R^{\uparrow }(H)\) might be empty; when \(n-k\) is odd, each element of \({\mathcal {G}}_{k-2}(V)\) is contained in at least one element of \(R^{\uparrow }(H)\). In the present paper, we investigate several properties of \(R^{\uparrow }(H)\), settle some open problems and propose a conjecture.     

Augmentation quotients for real representation rings of cyclic groups

Denote by \(C_m\) the cyclic group of order m. Let \({\mathcal {R}}(C_m)\) be its real representation ring, and \(\Delta (C_m)\) its augmentation ideal. In this paper, we give an explicit \({\mathbb {Z}}\)-basis for the n-th power \(\Delta ^{n}(C_m)\) and determine the isomorphism class of the n-th augmentation quotient \(\Delta ^n(C_m)/\Delta ^{n+1}(C_m)\) for each positive integer n.     

Tempered representations of <Emphasis Type="Italic">p</Emphasis>-adic groups: special idempotents and topology

Let \(G=G(k)\) be a connected reductive group over a p-adic field k. The smooth (and tempered) complex representations of G can be considered as the nondegenerate modules over the Hecke algebra \({\mathcal {H}}={\mathcal {H}}(G)\) and the Schwartz algebra \({\mathcal {S}}={\mathcal {S}}(G)\) forming abelian categories \({\mathcal {M}}(G)\) and \({\mathcal {M}}^t(G)\), respectively. Idempotents \(e\in {\mathcal {H}}\) or \({\mathcal {S}}\) define full subcategories \({\mathcal {M}}_e(G)= \{V : {\mathcal {H}}eV=V\}\) and \({\mathcal {M}}_e^t(G)= \{V : {\mathcal {S}}eV=V\}\). Such an e is said to be special (in \({\mathcal {H}}\) or \({\mathcal {S}}\)) if the corresponding subcategory is abelian. Parallel to Bernstein’s result for \(e\in {\mathcal {H}}\) we will prove that, for special \(e \in {\mathcal {S}}\), \({\mathcal {M}}_e^t(G) = \prod _{\Theta \in \theta _e} {\mathcal {M}}^t(\Theta )\) is a finite direct product of component categories \({\mathcal {M}}^t(\Theta )\), now referring to connected components of the center of \({\mathcal {S}}\). A special \(e\in {\mathcal {H}}\) will be also special in \({\mathcal {S}}\), but idempotents \(e\in {\mathcal {H}}\) not being special can become special in \({\mathcal {S}}\). To obtain conditions we consider the sets \(\mathrm{Irr}^t(G) \subset \mathrm{Irr}(G)\) of (tempered) smooth irreducible representations of G, and we view \(\mathrm{Irr}(G)\) as a topological space for the Jacobson topology defined by the algebra \({\mathcal {H}}\). We use this topology to introduce a preorder on the connected components of \(\mathrm{Irr}^t(G)\). Then we prove that, for an idempotent \(e \in {\mathcal {H}}\) which becomes special in \({\mathcal {S}}\), its support \(\theta _e\) must be saturated with respect to that preorder. We further analyze the above decomposition of \({\mathcal {M}}_e^t(G)\) in the case where G is k-split with connected center and where \(e = e_J \in {\mathcal {H}}\) is the Iwahori idempotent. Here we can use work of Kazhdan and Lusztig to relate our preorder on the support \(\theta _{e_J}\) to the reverse of the natural partial order on the unipotent classes in G. We finish by explicitly computing the case \(G=GL_n\), where \(\theta _{e_J}\) identifies with the set of partitions of n. Surprisingly our preorder (which is a partial order now) is strictly coarser than the reverse of the dominance order on partitions.     

Generalized <Emphasis Type="BoldItalic">r</Emphasis>-Lah numbers

In this paper, we consider a two-parameter polynomial generalization, denoted by \(\mathcal {G}_{a,b}(n,k;r)\), of the r-Lah numbers which reduces to these recently introduced numbers when a = b = 1. We present several identities for \(\mathcal {G}_{a,b}(n,k;r)\) that generalize earlier identities given for the r-Lah and r-Stirling numbers. We also provide combinatorial proofs of some earlier identities involving the r-Lah numbers by defining appropriate sign-changing involutions. Generalizing these arguments yields orthogonality-type relations that are satisfied by \(\mathcal {G}_{a,b}(n,k;r)\).     

Hypersurfaces with {H_{r+1}} = 0 in {\mathbb{H}^n \times \mathbb{R}}

We prove the existence of rotational hypersurfaces in \({\mathbb{H}^n \times \mathbb{R}}\) with \({H_{r+1} = 0}\) (r-minimal hupersurfaces) and we classify them. Then we prove some uniqueness theorems for r-minimal hypersurfaces with a given (finite or asymptotic) boundary. In particular, we obtain a Schoen-type theorem for two ended complete hypersurfaces.     

Codes with the identifiable parent property for multimedia fingerprinting

Let \({\mathcal {C}}\) be a q-ary code of length n and size M, and \({\mathcal {C}}(i) = \{\mathbf{c}(i) \ | \ \mathbf{c}=(\mathbf{c}(1), \mathbf{c}(2), \ldots , \mathbf{c}(n))^{T} \in {\mathcal {C}}\}\) be the set of ith coordinates of \({\mathcal {C}}\). The descendant code of a sub-code \({\mathcal {C}}^{'} \subseteq {\mathcal {C}}\) is defined to be \({\mathcal {C}}^{'}(1) \times {\mathcal {C}}^{'}(2) \times \cdots \times {\mathcal {C}}^{'}(n)\). In this paper, we introduce a multimedia analogue of codes with the identifiable parent property (IPP), called multimedia IPP codes or t-MIPPC(n, M, q), so that given the descendant code of any sub-code \({\mathcal {C}}^{'}\) of a multimedia t-IPP code \({\mathcal {C}}\), one can always identify, as IPP codes do in the generic digital scenario, at least one codeword in \({\mathcal {C}}^{'}\). We first derive a general upper bound on the size M of a multimedia t-IPP code, and then investigate multimedia 3-IPP codes in more detail. We characterize a multimedia 3-IPP code of length 2 in terms of a bipartite graph and a generalized packing, respectively. By means of these combinatorial characterizations, we further derive a tight upper bound on the size of a multimedia 3-IPP code of length 2, and construct several infinite families of (asymptotically) optimal multimedia 3-IPP codes of length 2.     

Generalizations of Schöbi’s Tetrahedral Dissection

Let v ₁,…,v _n be unit vectors in ? ⁿ such that v _i ?v _j =?w for i≠j, where \(-1. The points ∑ _i=1 ⁿ λ _i v _i (1≥λ ₁≥???≥λ _n ≥0) form a “Hill-simplex of the first type,” denoted by \(\mathcal {Q}_{n}(w)\). It was shown by Hadwiger in 1951 that \(\mathcal {Q}_{n}(w)\) is equidissectable with a cube. In 1985, Schöbi gave a three-piece dissection of \(\mathcal {Q}_{3}(w)\) into a triangular prism \(c\mathcal {Q}_{2}(\frac{1}{2})\times I\), where I denotes an interval and \(c=\sqrt{2(w+1)/3}\). In this paper, we generalize Schöbi’s dissection to an n-piece dissection of \(\mathcal {Q}_{n}(w)\) into a prism \(c\mathcal {Q}_{n-1}(\frac{1}{n-1})\times I\), where \(c=\sqrt{(n-1)(w+1)/n}\). Iterating this process leads to a dissection of \(\mathcal {Q}_{n}(w)\) into an n-dimensional rectangular parallelepiped (or “brick”) using at most n! pieces. The complexity of computing the map from \(\mathcal {Q}_{n}(w)\) to the brick is O(n ²). A second generalization of Schöbi’s dissection is given which applies specifically in ?⁴. The results have applications to source coding and to constant-weight binary codes.     

Resilience of ranks of higher inclusion matrices

Let \(n \ge r \ge s \ge 0\) be integers and \(\mathcal {F}\) a family of r-subsets of [n]. Let \(W_{r,s}^{\mathcal {F}}\) be the higher inclusion matrix of the subsets in \({{\mathcal {F}}}\) vs. the s-subsets of [n]. When \(\mathcal {F}\) consists of all r-subsets of [n], we shall simply write \(W_{r,s}\) in place of \(W_{r,s}^{\mathcal {F}}\). In this paper we prove that the rank of the higher inclusion matrix \(W_{r,s}\) over an arbitrary field K is resilient. That is, if the size of \(\mathcal {F}\) is “close” to \({n \atopwithdelims ()r}\) then \({{\mathrm{rank}}}_{K}( W_{r,s}^{\mathcal {F}}) = {{\mathrm{rank}}}_{K}(W_{r,s})\), where K is an arbitrary field. Furthermore, we prove that the rank (over a field K) of the higher inclusion matrix of r-subspaces vs. s-subspaces of an n-dimensional vector space over \({\mathbb {F}}_q\) is also resilient if \(\mathrm{char}(K)\) is coprime to q.     

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G _n /K _n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N _n nilpotent, in other words pairs (G _n , K _n ) for which G _n /K _n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L ²(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G _n /K _n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L ²(G/K). This allows an interpretation of L ²(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.     

Lebesgue weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-algebra on locally compact groups

Let G be a locally compact group, ω a weight function on G, and 1<p<∞. We introduce the Lebesgue weighted L ^p -space \({\mathcal{L}}_{\omega}^{1,p}(G)= L^{p}(G,\omega)\cap L^{1}(G)\) as a Banach space and introduce its dual. Furthermore, we consider this space as a Banach algebra with respect to the usual convolution and show that \({\mathcal{L}}_{\omega}^{1,p}(G)\) admits a bounded approximate identity if and only if G is discrete. In addition, we prove that amenability of this algebra implies that G is discrete and amenable. Moreover, we discuss the converse of this result.     

Fractional Differentiability for Solutions of Nonlinear Elliptic Equations

We study nonlinear elliptic equations in divergence form

$$\text {div }{\mathcal A}(x,Du)=\text {div } G.$$

When \({\mathcal A}\) has linear growth in D u, and assuming that \(x\mapsto {\mathcal A}(x,\xi )\) enjoys \(B^{\alpha }_{\frac {n}\alpha , q}\) smoothness, local well-posedness is found in \(B^{\alpha }_{p,q}\) for certain values of \(p\in [2,\frac {n}{\alpha })\) and \(q\in [1,\infty ]\). In the particular case \({\mathcal A}(x,\xi )=A(x)\xi \), G = 0 and \(A\in B^{\alpha }_{\frac {n}\alpha ,q}\), \(1\leq q\leq \infty \), we obtain \(Du\in B^{\alpha }_{p,q}\) for each \(p<\frac {n}\alpha \). Our main tool in the proof is a more general result, that holds also if \({\mathcal A}\) has growth s?1 in D u, 2 ≤ s ≤ n, and asserts local well-posedness in L ^q for each q > s, provided that \(x\mapsto {\mathcal A}(x,\xi )\) satisfies a locally uniform VMO condition.

     

Relative Cartier divisors and Laurent polynomial extensions

If \(i:A\subset B\) is a commutative ring extension, we show that the group \({\mathcal I}(A,B)\) of invertible A-submodules of B is contracted in the sense of Bass, with \(L{\mathcal I}(A,B)=H^0_{\mathrm {et}}(A,i_*{\mathbb Z}/{\mathbb Z})\). This gives a canonical decomposition for \({\mathcal I}(A[t,\frac{1}{t}],B[t,\frac{1}{t}])\).     

On Irreducible Representations of the Zassenhaus Superalgebras with p-Characters of Height 0

Let n be a positive integer, and \(\mathfrak {A}(n)=\mathbb {F}[x]/(x^{p^{n}})\), the divided power algebra over an algebraically closed field \(\mathbb {F}\) of prime characteristic p >?2. Let π(n) be the tensor product of \(\mathfrak {A}(n)\) and the Grassmann superalgebra \(\bigwedge (1)\) in one variable. The Zassenhaus superalgebra \(\mathcal {Z}(n)\) is defined to be the Lie superalgebra of the special super derivations of the superalgebra π(n). In this paper we study simple modules over the Zassenhaus superalgebra \(\mathcal {Z}(n)\) with p-characters of height 0. We give a complete classification of the isomorphism classes of such simple modules and determine their dimensions. A sufficient and necessary condition for the irreducibility of Kac modules is obtained.     

EKR Sets for Large <Emphasis Type="Italic">n</Emphasis> and <Emphasis Type="Italic">r</Emphasis>

Let \(\mathcal {A}\subset \left( {\begin{array}{c}[n]\\ r\end{array}}\right) \) be a compressed, intersecting family and let \(X\subset [n]\). Let \(\mathcal {A}(X)=\{A\in \mathcal {A}:A\cap X\ne \emptyset \}\) and \(\mathcal {S}_{n,r}=\left( {\begin{array}{c}[n]\\ r\end{array}}\right) (\{1\})\). Motivated by the Erd?s–Ko–Rado theorem, Borg asked for which \(X\subset [2,n]\) do we have \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\) for all compressed, intersecting families \(\mathcal {A}\)? We call X that satisfy this property EKR. Borg classified EKR sets X such that \(|X|\ge r\). Barber classified X, with \(|X|\le r\), such that X is EKR for sufficiently large n, and asked how large n must be. We prove n is sufficiently large when n grows quadratically in r. In the case where \(\mathcal {A}\) has a maximal element, we sharpen this bound to \(n>\varphi ^{2}r\) implies \(|\mathcal {A}(X)|\le |\mathcal {S}_{n,r}(X)|\). We conclude by giving a generating function that speeds up computation of \(|\mathcal {A}(X)|\) in comparison with the naïve methods.     

Cyclotomic Hecke Algebras of <Emphasis Type="Italic">G</Emphasis>(<Emphasis Type="Italic">r</Emphasis>, <Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)

In this note, we find a monomial basis of the cyclotomic Hecke algebra \({\mathcal{H}_{r,p,n}}\) of G(r,p,n) and show that the Ariki-Koike algebra \({\mathcal{H}_{r,n}}\) is a free module over \({\mathcal{H}_{r,p,n}}\), using the Gröbner-Shirshov basis theory. For each irreducible representation of \({\mathcal{H}_{r,p,n}}\), we give a polynomial basis consisting of linear combinations of the monomials corresponding to cozy tableaux of a given shape.     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

Some infinite families of Ramsey ($${\varvec{P}}_\mathbf{3},{\varvec{P}}_{{\varvec{n}}}$$)-minimal trees

For any given two graphs G and H, the notation \(F\rightarrow \) (G, H) means that for any red–blue coloring of all the edges of F will create either a red subgraph isomorphic to G or a blue subgraph isomorphic to H. A graph F is a Ramsey (G, H)-minimal graph if \(F\rightarrow \) (G, H) but \(F-e\nrightarrow (G,H)\), for every \(e \in E(F)\). The class of all Ramsey (G, H)-minimal graphs is denoted by \(\mathcal {R}(G,H)\). In this paper, we construct some infinite families of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n=8\) and 9. In particular, we give an algorithm to obtain an infinite family of trees belonging to \(\mathcal {R}(P_3,P_n)\), for \(n\ge 10\).     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Holomorphic invariant forms of a bounded domain

Given a complete ortho-normal system  = (0, 1, 2, . . .) of L2H(D), the space of holomorphic and absolutely square integrable functions in the bounded domain D of Cn, we construct a holomorphic imbedding ι : D →■(n, ∞), the complex infinite dimensional Grassmann manifold of rank n. It is known that in ■(n, ∞) there are n closed (μ, μ)-forms τμ (μ = 1, . . . , n) which are invariant under the holomorphic isometric automorphism of ■(n, ∞) and generate algebraically all the harmonic differential forms of ...     

On Multiple and Infinite Log-Concavity

Following Boros-Moll, a sequence (a _n ) is m-log-concave if \({\mathcal{L}^{j}(a_{n})\geqslant0}\) for all j =  0, 1, . . . , m. Here, \({\mathcal{L}}\) is the operator defined by \({\mathcal{L}(a_{n}) = a^{2}_{n}-a_{n-1}a_{n+1}}\). By a criterion of Craven-Csordas and McNamara-Sagan it is known that a sequence is ∞-log-concave if it satisfies the stronger inequality \({a^{2}_{k}\geqslant ra_{k-1}a_{k+1}}\) for large enough r. On the other hand, a recent result of Brändén shows that ∞-log-concave sequences include sequences whose generating polynomial has only negative real roots. In this paper, we investigate sequences which are fixed by a power of the operator \({\mathcal{L}}\) and are therefore ∞-log-concave for a very different reason. Surprisingly, we find that sequences fixed by the non-linear operators \({\mathcal{L}}\) and \({\mathcal{L}^{2}}\) are, in fact, characterized by a linear 4-term recurrence. In a final conjectural part, we observe that positive sequences appear to become ∞-log-concave if convoluted with themselves a finite number of times.