The long root geometry A n , { 1 , n } ( 𝕂 ) for the special linear group SL ( n + 1 , 𝕂 ) admits an embedding in the (projective space of) the vector space of the traceless square matrices of order n + 1 with entries in the field 𝕂 , usually regarded as the natural embedding of A n , { 1 , n } ( 𝕂 ) . S. Smith and H. Völklein in [10] have proved that the natural embedding of A 2 , { 1 , 2 } ( 𝕂 ) is relatively universal if and only if 𝕂 is either algebraic over its minimal subfield or perfect with positive characteristic. They also give some information on the relatively universal embedding of A 2 , { 1 , 2 } ( 𝕂 ) which covers the natural one, but that information is not sufficient to exhaustively describe it. The “if” part of Smith-Völklein’s result also holds true for any n , as proved by Völklein in [13] in his investigation of the adjoint modules of Chevalley groups. In this paper we give an explicit description of the relatively universal embedding of A n , { 1 , n } ( 𝕂 ) which covers the natural one. In particular, we prove that this relatively universal embedding has (vector) dimension equal to 𝔡 + n 2 + 2 n where 𝔡 is the transcendence degree of 𝕂 over its minimal subfield (if char ( 𝕂 ) = 0 ) or the generating rank of 𝕂 over 𝕂 p (if char ( 𝕂 ) = p > 0 ). Accordingly, both the “if” and the “only if” part of Smith-Völklein’s result hold true for every n ≥ 2 .

We prove that the existence of a divisor of degree 3 and Baker-Norine rank at least 1 on a 3 -edge connected tropical curve is equivalent to the existence of a non-degenerate harmonic morphism of degree 3 from a tropical modification of it to a tropical rational curve. Using the second description, we define the moduli spaces of 3 -edge connected tropical trigonal covers and of 3 -edge connected tropical trigonal curves, the latter as a locus in the moduli space of tropical curves. Finally, we prove that the moduli space of 3 -edge connected genus g tropical trigonal curves has the same dimension as the moduli space of genus g algebraic trigonal curves.

In a 2018 paper, Davis and Sagan studied several pattern-avoiding polytopes. They found that a particular pattern-avoiding Birkhoff polytope had the same normalized volume as the order polytope of a certain poset, leading them to ask if the two polytopes were unimodularly equivalent. Motivated by Davis and Sagan’s question, in this paper we define a pattern-avoiding Birkhoff polytope called a c -Birkhoff polytope for each Coxeter element c of the symmetric group. We then show that the c -Birkhoff polytope is unimodularly equivalent to the order polytope of the heap poset of the c -sorting word of the longest permutation. When c = s 1 s 2 ⋯ s n , this result recovers an affirmative answer to Davis and Sagan’s question. Another consequence of this result is that the normalized volume of the c -Birkhoff polytope is the number of the longest chains in the (type A) c -Cambrian lattice.

We prove necessary conditions for certain elementary symmetric functions, e λ , to appear with nonzero coefficient in Stanley’s chromatic symmetric function as well as in the generalization considered by Shareshian and Wachs. We do this by first considering the expansion in the monomial or Schur basis and then performing a basis change. Using the former, we make a connection with two fundamental graph theory invariants, the independence and clique numbers. This allows us to prove nonnegativity of three-column coefficients for all natural unit interval graphs, giving more insight into the Stanley–Stembridge Conjecture, recently proven by Hikita, and the Shareshian–Wachs Conjecture. The Schur basis permits us to give a new interpretation of the coefficient of e n in terms of tableaux. We are also able to give an explicit formula for that coefficient.

We prove the switching equivalence of the strongly regular polar graphs N O ± ( 4 m , 2 ) , N O ∓ ( 2 m + 1 , 4 ) , and Γ ( O ∓ ( 4 m , 2 ) ) with an isolated vertex by giving an analytic description for them and their associated two-graphs.

A quasigroup is a pair ( Q , * ) where Q is a non-empty set and * is a binary operation on Q such that for every ( u , v ) ∈ Q 2 there exists a unique ( x , y ) ∈ Q 2 such that u * x = v = y * u . Let q be an odd prime power, let 𝔽 q denote the finite field of order q , and let ℛ q denote the set of non-zero squares in 𝔽 q . Let ( a , b ) ∈ 𝔽 q 2 be such that { a b , ( a - 1 ) ( b - 1 ) } ⊆ ℛ q . Let 𝒬 a , b denote the quadratic quasigroup ( 𝔽 q , * a , b ) where * a , b is defined by x * a , b y = x + a ( y - x ) if y - x ∈ ℛ q , x + b ( y - x ) otherwise . The operation table of a quadratic quasigroup is a quadratic Latin square. Recently, it has been determined exactly when two quadratic quasigroups are isomorphic and the automorphism group of any quadratic quasigroup has been determined. In this paper, we extend these results. We determine exactly when two quadratic quasigroups are isotopic and we determine the autotopism group of any quadratic quasigroup. In the process, we count the number of 2 × 2 subsquares in quadratic Latin squares.

Generalizing the notion of a multiplicative inequality among minors of a totally positive matrix, we describe, over full rank cluster algebras of finite type, the cone of Laurent monomials in cluster variables that are bounded as real-valued functions on the positive locus of the cluster variety. We prove that the extreme rays of this cone are the u -variables of the cluster algebra. Using this description, we prove that all bounded ratios are bounded by 1 and give a sufficient condition for all such ratios to be subtraction free. This allows us to show in Gr ( 2 , n ) , Gr ( 3 , 6 ) , Gr ( 3 , 7 ) , and Gr ( 3 , 8 ) that every bounded Laurent monomial in Plücker coordinates factors into a positive integer combination of so-called primitive ratios. In Gr ( 4 , 8 ) this factorization does not exists, but we provide the full list of extreme rays of the cone of bounded Laurent monomials in Plücker coordinates.

We provide a formula for the Garsia-Remmel q -rook numbers as a sum over standard Young tableaux. We connect our formula with the coefficients in q -Whittaker expansion of unicellular LLT functions.

We study real and positive tropicalizations of the varieties of low rank symmetric matrices over real or complex Puiseux series. We show that real tropicalization coincides with complex tropicalization for rank two and corank one cases. We also show that the two notions of positive tropicalization introduced by Speyer and Williams coincide for symmetric rank two matrices, but they differ for symmetric corank one matrices.

Let G be a finite simple graph and let ind-match ( G ) and ord-match ( G ) denote the induced matching number and the ordered matching number of G , respectively. We characterize all bipartite graphs G with ind-match ( G ) = ord-match ( G ) . We establish the Castelnuovo-Mumford regularity of powers of edge ideals and depth of powers of cover ideals for such graphs.