Patrias and Pylyavskyy introduced shifted Hecke insertion as an application of their theory of dual filtered graphs. We use shifted Hecke insertion to construct symmetric function representatives for the K-theory of the orthogonal Grassmannian. These representatives are closely related to the shifted Grothendieck polynomials of Ikeda and Naruse. We then recover the K-theory structure coefficients of Clifford-Thomas-Yong/Buch-Samuel by introducing a shifted K-theoretic Poirier-Reutenauer algebra. Our proofs depend on the theory of shifted K-theoretic jeu de taquin and the weak K-Knuth relations.1 representatives for the Grassmannian. We will see in Proposition 3.5 that the K λ and the GP λ of Ikeda and Naruse are closely related. As a corollary, we derive a new proof of the symmetry of GP λ . Surprisingly, while GP λ satisfies the Q-cancellation property, we show that K λ does not, hence cannot be expressed as a sum of Schur P -functions.Using weak K-Knuth equivalence, we define a shifted K-theoretic Poirier-Reutenauer algebra. This generalizes the shifted Poirier-Reutenauer algebra of Jang and Li [10], a shifted analogue of the Poirier-Reutenauer Hopf algebra [14]. Our approach follows work of Patrias and Pylyavskyy on a K-theoretic Poirier-Reutenauer bialgebra [13]. The shifted K-theoretic Poirier-Reutenauer algebra does not have the coalgebra structure analogous to that of shifted Poirier-Reutenauer and K-theoretic Poirier Reutenauer. Using the shifted Ktheoretic Poirier-Reutenauer algebra, we define a Littlewood-Richardson rule for the product K λ · K µ . This rule coincides up to sign with the rule of Clifford, Thomas, and Yong [5] and Buch and Samuel [4]. As a consequence, we have our main result.The remainder of the paper is structured as follows. In the next section, we define shifted Hecke insertion and show how it relates to shifted K-theoretic jeu de taquin. This allows us to demonstrate the relationship between shifted Hecke insertion and the weak K-Knuth equivalence relations. The third section is devoted to defining the K λ , expressing them in terms of shifted Hecke insertion and proving Theorem 1.1. In the fourth and final section, we develop the shifted K-theoretic Poirier-Reutenauer algebra and prove Theorem 1.2. Shifted Hecke Insertion and Weak K-Knuth EquivalenceWe show that the shifted Hecke insertion given in [12] respects the weak K-Knuth equivalence given in [4]. Before presenting our argument, we review previous work on increasing shifted tableaux, shifted Hecke insertion, and shifted K-jeu de taquin.2.1. Increasing shifted tableaux. To each strict partition λ = (λ 1 > λ 2 > . . . > λ k ) we associate the shifted shape, which is an array of boxes where the ith row has λ i boxes and is indented i − 1 units. A shifted tableau is a filling of the shifted shape with positive integers. A shifted tableau is increasing if the labels are strictly increasing from left to right along rows and top to bottom down columns. The reading word of an increasing shifted tableau T , denoted row(T ), is the w...

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Abstract. We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Lê and Zagier.

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A large family of relations among multiple zeta values may be described using the combinatorics of shuffle and quasi-shuffle algebras. While the structure of shuffle algebras have been well understood for some time now, quasi-shuffle algebras were only formally studied relatively recently. In particular, Hoffman [10] gives a thorough discussion of the algebraic structure, including a choice of algebra basis, and applies his results to produce families of relations among multiple zeta values and their generalisations [11]. In a series of recent talks, Hirose and Sato proposed a conjectural family of relations coming from a new generalised shuffle structure, lifting a set of graded relations established by the author [13] to conjectural genuine relations. In this paper, we define a commutative algebra structure on the space of non-commutative polynomials in a countable alphabet, generalising the shuffle-like structure of Hirose and Sato. We show that, over the rational numbers, this generalised quasi-shuffle algebra is isomorphic to the standard shuffle algebra, allowing us to reproduce most of Hoffman's results on quasi-shuffle algebras. We then apply these results to the case of multiple zeta values, reproducing several known families of results and suggesting several more which can be verified numerically, hence providing further evidence for the conjecture of Hirose and Sato.