Marie-Charlotte Brandenburg, Georg Loho, Rainer Sinn
We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.
{"title":"Tropical positivity and determinantal varieties","authors":"Marie-Charlotte Brandenburg, Georg Loho, Rainer Sinn","doi":"10.5802/alco.286","DOIUrl":"https://doi.org/10.5802/alco.286","url":null,"abstract":"We initiate the study of positive-tropical generators as positive analogues of the concept of tropical bases. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. We focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136248834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The quasisymmetric Macdonald polynomials G γ (X;q,t) were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials P λ (X;q,t) with the property that G γ (X;0,0) equals QS γ (X), the quasisymmetric Schur polynomial of [9]. We derive an expansion for G γ (X;q,t) in the fundamental basis of quasisymmetric functions.
{"title":"Expanding the quasisymmetric Macdonald polynomials in the fundamental basis","authors":"Sylvie Corteel, Olya Mandelshtam, Austin Roberts","doi":"10.5802/alco.289","DOIUrl":"https://doi.org/10.5802/alco.289","url":null,"abstract":"The quasisymmetric Macdonald polynomials G γ (X;q,t) were recently introduced by the first and second authors with Haglund, Mason, and Williams in [3] to refine the symmetric Macdonald polynomials P λ (X;q,t) with the property that G γ (X;0,0) equals QS γ (X), the quasisymmetric Schur polynomial of [9]. We derive an expansion for G γ (X;q,t) in the fundamental basis of quasisymmetric functions.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136242788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper concerns the symmetric and anti-symmetric Kronecker products of characters of the symmetric groups. We provide new closed formulas for decomposing these products, unexpected connections with 2-modular decomposition numbers, Catalan combinatorics, and a refinement of the famous Saxl conjecture.
{"title":"Splitting Kronecker squares, 2-decomposition numbers, Catalan combinatorics, and the Saxl conjecture","authors":"Christine Bessenrodt, Chris Bowman","doi":"10.5802/alco.294","DOIUrl":"https://doi.org/10.5802/alco.294","url":null,"abstract":"This paper concerns the symmetric and anti-symmetric Kronecker products of characters of the symmetric groups. We provide new closed formulas for decomposing these products, unexpected connections with 2-modular decomposition numbers, Catalan combinatorics, and a refinement of the famous Saxl conjecture.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136280204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A q-analog of the adjacency matrix of the n-cube","authors":"Subhajit Ghosh, Murali Srinivasan","doi":"10.5802/alco.282","DOIUrl":"https://doi.org/10.5802/alco.282","url":null,"abstract":"","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42648303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2 d -dimensional triangulations without interior ( d + 1)- simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d -simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.
{"title":"Quiver combinatorics and triangulations of cyclic polytopes","authors":"Nicholas J. Williams","doi":"10.5802/alco.280","DOIUrl":"https://doi.org/10.5802/alco.280","url":null,"abstract":"Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2 d -dimensional triangulations without interior ( d + 1)- simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d -simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48334817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its vertex-Cheeger constant. We also prove an analogous result for a large class of irregular graphs, obtained as spanning subgraphs of vertex-transitive graphs. Using a result of Babai, we obtain a lower bound for the smallest eigenvalue of the normalised adjacency operator of a vertex-transitive graph in terms of its diameter and its degree.
{"title":"A spectral bound for vertex-transitive graphs and their spanning subgraphs","authors":"Arindam Biswas, Jyoti Prakash Saha","doi":"10.5802/alco.278","DOIUrl":"https://doi.org/10.5802/alco.278","url":null,"abstract":"For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its vertex-Cheeger constant. We also prove an analogous result for a large class of irregular graphs, obtained as spanning subgraphs of vertex-transitive graphs. Using a result of Babai, we obtain a lower bound for the smallest eigenvalue of the normalised adjacency operator of a vertex-transitive graph in terms of its diameter and its degree.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135336216","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let f ( z ) = P ∞ n =0 ( − 1) n z n /n ! n !. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1 /f ( z ) = P ∞ n =0 ω n z n /n ! n !. They proved that ω n counts the number of pairs of permutations of the n th symmetric group S n with no common ascent. This paper gives a combinatorial interpretation of a natural q -analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n ( q ) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: P n i =0 ( − 1) i e i h n − i = 0, which then generalizes our q -analogue to a symmetric group representation result.
让f (z) = P∞n = 0 n (n−1)z / n !n !。在1975年的这篇文章,Carlitz是97万,沃恩provided a combinatorial境coefficients电源的解析》系列1 - f (z) = z P∞n = 0ωn n / n !n !。他们proved thatωn算数of副of permutations当家》n th symmetric集团S n与普通ascent号。这篇文章给a combinatorial解释of a自然q -analogue of top homology》ωn: studying Segre广告《眼泪lattice B n (q)和不由自主。我们也derive an equation就是analogous to a well-known symmetric功能身份:P n i = 0(−1)我e h n−i = 0,哪种然后我们generalizes q -analogue to a symmetric集团representation论点。
{"title":"A q-analogue of a result of Carlitz, Scoville and Vaughan via the homology of posets","authors":"Yifei Li","doi":"10.5802/alco.265","DOIUrl":"https://doi.org/10.5802/alco.265","url":null,"abstract":"Let f ( z ) = P ∞ n =0 ( − 1) n z n /n ! n !. In their 1975 paper, Carlitz, Scoville and Vaughan provided a combinatorial interpretation of the coefficients in the power series 1 /f ( z ) = P ∞ n =0 ω n z n /n ! n !. They proved that ω n counts the number of pairs of permutations of the n th symmetric group S n with no common ascent. This paper gives a combinatorial interpretation of a natural q -analogue of ω n by studying the top homology of the Segre product of the subspace lattice B n ( q ) with itself. We also derive an equation that is analogous to a well-known symmetric function identity: P n i =0 ( − 1) i e i h n − i = 0, which then generalizes our q -analogue to a symmetric group representation result.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43945283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatorial interpretation of the Kronecker coefficients. As avenues for future research, we discuss applications of the ribbon graph quantum mechanics in algorithms for quantum computation. We also describe a quantum membrane interpretation of these quantum mechanical systems.
{"title":"Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients","authors":"Joseph Ben Geloun, Sanjaye Ramgoolam","doi":"10.5802/alco.254","DOIUrl":"https://doi.org/10.5802/alco.254","url":null,"abstract":"We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatorial interpretation of the Kronecker coefficients. As avenues for future research, we discuss applications of the ribbon graph quantum mechanics in algorithms for quantum computation. We also describe a quantum membrane interpretation of these quantum mechanical systems.","PeriodicalId":36046,"journal":{"name":"Algebraic Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134923114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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